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# A group-theoretical approach for nonlinear Schrödinger equations

Giovanni Molica Bisci
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• Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari – Feo di Vito, 89124 Reggio Calabria, Italy
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Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/acv-2018-0016

## Abstract

The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space ${ℝ}^{d}$ ($d\ge 3$). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of the orthogonal group $O\left(d\right)$. In addition, if the nonlinear term is odd, and $d>3$, the existence of ${\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$ pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the ${L}^{\mathrm{\infty }}$-positive and radially symmetric potential term W governing the Schrödinger equations is requested. A concrete example of an application is pointed out. Finally, we emphasize that the method adopted here should be applied for a wider class of energies largely studied in the current literature also in non-Euclidean setting as, for instance, concave-convex nonlinearities on Cartan–Hadamard manifolds with poles.

MSC 2010: 35J91; 35J60; 35A01; 45A15

## 1 Introduction

The aim of this paper is to establish an existence result for the following Schrödinger equation:

(Slambda)

where $\left({ℝ}^{d},|\cdot |\right)$ is the Euclidean space (with $d\ge 3$), $\mathrm{\Delta }u:=div\left(\nabla u\right)$ stands for the classical Laplacian operator, $f:ℝ\to ℝ$ is a subcritical continuous function, $W\in {L}^{\mathrm{\infty }}\left({ℝ}^{N}\right)\cap {L}^{2}\left({ℝ}^{d}\right)\setminus \left\{0\right\}$ is a non-negative radially symmetric map and λ is a positive real parameter. Finally, in order to avoid technicalities, we assume that the potential $V:{ℝ}^{d}\to ℝ$ satisfies the following condition:

• ${(h_{V}^{d})}$

$V\in {C}^{0}\left({ℝ}^{d}\right)$ is radially symmetric and ${inf}_{x\in {ℝ}^{d}}V\left(x\right)>0$.

As is well known, the interest in Schrödinger equations comes from various problems in mathematical physics, among others: cosmology, constructive field theory, solitary waves, and nonlinear Klein–Gordon equations. See, for instance, the quoted papers [4, 18, 40].

Moreover, a solution of problem (Slambda) can also be interpreted as a stationary state of the reaction diffusion equation

${u}_{t}=\mathrm{\Delta }u-V\left(x\right)u+\lambda W\left(x\right)f\left(u\right),$

that, as pointed out in [13], describe some chemical phenomena (see [21] for details).

In quantum mechanics, the Schrödinger equation modelled the motion of microscopic particles. The nonlinear term represents the interaction of two particles. This interaction mainly acts as attraction when the energy of the particles is small. A stable state is described thought the existence of non-trivial solutions.

Of course, we do not intend to review the huge bibliography of equations like (Slambda). We just emphasize that the potential V has a crucial role concerning the existence and behavior of solutions. For instance, after the seminal paper of Rabinowitz [37] where the potential V is assumed to be coercive, several different assumptions are adopted in order to obtain existence and multiplicity results (see, among others, the papers [2, 13, 11, 10, 23]). When V is a positive constant, or V is radially symmetric, it is natural to look for radially symmetric solutions, see [40, 42].

From a purely mathematical point of view, it is worth mentioning that the early classical studies dedicated to equation (Slambda) were given, among others, by Strauss [40], Bartsch and Willem [14], Berestycki and Lions [15], and Struwe [41]. An important incentive to its study was provided by the work [24] of Gidas, Ni and Nirenberg, where a functional analysis approach was proposed to attack it. We cite, amid the wide literature on the subject, the works [13, 14] where Schrödinger equations were studied by exploiting different methods and new technical approaches.

The main difficulty in dealing with problem like (Slambda) arises from the lack of compactness, which becomes clear when one looks at the following case:

(L)

The set of solutions of (L) is invariant under translations and, therefore, it is not compact. If both V and W are radially symmetric, then compactness can be restored by looking only for radially symmetric solutions. For instance, the existence of infinitely many radially symmetric solutions has been established, among others, by Bartsch and Willem [14], Berestycki and Lions [15], and Conti, Merizzi and Terracini [19].

The existence of non-radial solutions (bound states) of problem (Slambda) or some of its variants has been studied after the paper of Bartsch and Willem [13] (see, for instance, Willem’s book [42] and the closed related papers by Kristály [28] and Kristály, Moroşanu and O’Regan [29]). We also cite the paper [17] of Clapp and Weth, and the references therein, where the authors under some weak onesided asymptotic estimates on the terms V and W (without symmetry assumptions), proved the existence of at least $\frac{d}{2}+1$ pairs of non-trivial weak solutions for suitable Schrödinger equations; see, for completeness, also the related papers given by Bartsch, Clapp and Weth [9], and Evéquoz and Weth [20].

In the present paper we find non-radial solutions of problem (Slambda) for $d=4$ or $d\ge 6$ provided W and f satisfy respectively a summability condition and a growth assumption and f is odd. Following [14], an interesting prototype for problem (Slambda) is given by

where $1, the potential V is bounded below by a positive constant and W is bounded.

In this setting, a simple case of our main result reads as follows.

#### Theorem 1.

Let $d\mathrm{>}\mathrm{3}$ and let $\mathrm{1}\mathrm{<}r\mathrm{<}\mathrm{2}\mathrm{<}s\mathrm{<}{\mathrm{2}}^{\mathrm{*}}$, where ${\mathrm{2}}^{\mathrm{*}}\mathrm{:=}\frac{\mathrm{2}\mathit{}d}{d\mathrm{-}\mathrm{2}}$. Furthermore, let t be the conjugate exponent of s and let $W\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\cap }{L}^{t}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$ be a non-negative radially symmetric map. Finally, let V be a potential for which $\mathrm{\left(}{h}_{V}^{d}\mathrm{\right)}$ holds. Then, for λ sufficiently small, the problem

(Clambda)

${\zeta }_{S}^{\left(d\right)}:=1+{\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of non-trivial weak solutions ${\mathrm{\left\{}\mathrm{±}{u}_{\lambda \mathrm{,}i}\mathrm{\right\}}}_{i\mathrm{\in }{J}^{\mathrm{\prime }}}\mathrm{\subset }{E}_{V}$, where ${J}^{\mathrm{\prime }}\mathrm{:=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}{\zeta }_{S}^{\mathrm{\left(}d\mathrm{\right)}}\mathrm{\right\}}$, such that

$\underset{\lambda \to {0}^{+}}{lim}\left({\int }_{{ℝ}^{d}}{|\nabla {u}_{\lambda ,i}\left(x\right)|}^{2}𝑑x+{\int }_{{ℝ}^{d}}V\left(x\right){|{u}_{\lambda ,i}\left(x\right)|}^{2}𝑑x\right)=0,$

and $\mathrm{|}{u}_{\lambda \mathrm{,}i}\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{|}\mathrm{\to }\mathrm{0}$, as $\mathrm{|}x\mathrm{|}\mathrm{\to }\mathrm{\infty }$, for every $i\mathrm{\in }{J}^{\mathrm{\prime }}$. Moreover, if $d\mathrm{\ne }\mathrm{5}$, problem (Clambda) admits at least

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of sign-changing weak solutions ${\mathrm{\left\{}\mathrm{±}{z}_{\lambda \mathrm{,}i}\mathrm{\right\}}}_{i\mathrm{\in }J}\mathrm{\subset }{E}_{V}$, where $J\mathrm{:=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}{\tau }_{d}\mathrm{\right\}}$.

Hereafter, $\left[x\right]$ denotes the integer part of a real number $x\ge 0$.

More precisely, in Theorem 6 we prove that there exists a well-localized interval of positive real parameters $\left(0,{\lambda }^{\star }\right)$ such that, for every $\lambda \in \left(0,{\lambda }^{\star }\right)$, problem (Slambda) admits at least one non-trivial radial weak solution in a suitable Sobolev space ${E}_{V}$. By a Strauss-type estimate (see Lions [40]) every weak solution ${u}_{\lambda }\in {E}_{V}$ is also homoclinic, that is,

The proof of Theorem 6 is based on variational methods. We find critical points of the energy functional

${J}_{\lambda }\left(u\right):=\frac{1}{2}\left({\int }_{{ℝ}^{d}}{|\nabla u\left(x\right)|}^{2}𝑑x+{\int }_{{ℝ}^{d}}V\left(x\right){|u\left(x\right)|}^{2}𝑑x\right)-\lambda {\int }_{{ℝ}^{d}}W\left(x\right)\left({\int }_{0}^{u\left(x\right)}f\left(z\right)𝑑z\right)𝑑x$

for every $u\in {E}_{V}$, associated with problem (Slambda) by means of a local minimum result for differentiable functionals (see Theorem 3) and the well-known Palais’ principle of symmetric criticality (see Theorem 4).

A key assumption in Theorem 6 is given by

$\underset{t\to {0}^{+}}{lim sup}\frac{{\int }_{0}^{t}f\left(z\right)𝑑z}{{t}^{2}}=+\mathrm{\infty },\text{and}\mathit{ }\underset{t\to {0}^{+}}{lim inf}\frac{{\int }_{0}^{t}f\left(z\right)𝑑z}{{t}^{2}}>-\mathrm{\infty },$

see Remark 8. This condition is not novel in the current literature and it has been used in order to study existence and multiplicity results for some classes of elliptic problems on bounded domains (see, among others, the papers of Kajikiya [27, 26, 25] and Molica Bisci and Servadei [35]). To the best of our knowledge, no previous applications to Schrödinger equations have been achieved requiring this hypothesis.

Successively, in Theorem 11, the existence of multiple solutions with no radial symmetry for problem (Slambda) is studied. Under the additional hypotheses of oddness on f we observe that the energy functional ${J}_{\lambda }$ is invariant under certain subgroups actions of the orthogonal group $O\left(d\right)$. Due to Theorem 3, every restriction of ${J}_{\lambda }$ to the appropriate subspaces of invariant functions admit a non-trivial critical point. Thanks to the principle of symmetric criticality of Palais, these points will be also critical points of the original functional ${J}_{\lambda }$, and they are non-radial solutions of problem (Slambda), depending on the choice of the subgroup of $O\left(d\right)$; see the related paper [28].

More precisely, thanks to the above careful group-theoretical analysis inspired by Bartsch and Willem [13] and developed by Kristály, Moroşanu and O’Regan in [29], we construct

${\zeta }_{S}^{\left(d\right)}:=1+{\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

subspaces of ${E}_{V}$. Moreover, when $d\ne 5$, there are

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

of this subspaces that does not contain radial symmetric functions; see [13] and [29, Theorem 2.2]. Further energy-level analysis together the local minimum result recalled in Theorem 3 provides at least one pair of non-trivial weak solutions for problem (Slambda) belonging to these subspaces separately whenever the parameter λ is sufficiently small, that is,

where ${\lambda }^{*}$ and ${\lambda }_{i,q}^{\star }$ are suitable geometrical constants whose expression is given respectively in formulas (3.2) and (4.4). The structure of the parameters ${\lambda }^{\star }$ and ${\lambda }_{\star }$ displayed in Theorems 6 and 11 is not simple and their value depends on certain Sobolev embedding constants and on the datum f.

We notice that in [29, Theorem 1.1] the authors proved some (non-)existence and multiplicity results for possible perturbed Schrödinger equations. Their existence and multiplicity results are valid for every λ sufficiently large by requiring a suitable behavior of the nonlinear term f at zero and at infinity. It is easily seen that [29, Theorem 1.1] cannot be applied to problem (Plambda) studied in Example 15. On the other hand, their non-existence result ensures that problem (Slambda) has only the trivial (identically zero) solution for λ sufficiently small requiring, among others assumptions, the following asymptotic behavior at zero:

$\underset{z\to 0}{lim}\frac{f\left(z\right)}{z}=0.$

The above condition clearly fails dealing with a nonlinear concave-convex term like

where $1, as is given in Example 15 (see Remark 16).

Let us also note that Theorems 6 and 11 might actually be reformulated for a wider class of elliptic problems by using our technical approach. For instance, possible extensions to variational-hemivariational inequalities as well as applications to Cartan–Hadamard manifolds (i.e. Riemannian manifolds that are complete, simply-connected, and with non-positive sectional curvature) and non-local fractional problems (see [34]) will be examined in some forthcoming papers.

The manuscript is organized as follows. In Section 2 we give some notations and we recall some properties of the functional space we work in. In order to apply critical point methods to problem (Slambda), we need to work in a subspace of the functional space ${E}_{V}$; in particular, we give some tools which will be useful along the paper (see Propositions 2 and Lemma 5). In Section 3 we study problem (Slambda) and we prove our existence result (see Theorem 6). Finally, in the last section we study the existence of multiple non-radial solutions to the problem (Slambda) for λ sufficiently small.

We cite the books [3, 1, 30, 42] as general references on the subject treated along the paper. The bibliography does no escape the usual rule being incomplete.

## 2 Abstract framework

To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. Assume $d\ge 3$ and let

${E}_{V}:=\left\{u\in {H}^{1}\left({ℝ}^{d}\right):{\int }_{{ℝ}^{d}}\left({|\nabla u\left(x\right)|}^{2}+V\left(x\right){|u\left(x\right)|}^{2}\right)𝑑x<+\mathrm{\infty }\right\}$

be the Hilbert space endowed by the inner product

and induced norm

${\parallel u\parallel }_{{E}_{V}}:={\left({\int }_{{ℝ}^{d}}{|\nabla u\left(x\right)|}^{2}𝑑x+{\int }_{{ℝ}^{d}}V\left(x\right){|u\left(x\right)|}^{2}𝑑x\right)}^{\frac{1}{2}}$

for every $u\in {E}_{V}$.

Thanks to assumption $\left({h}_{V}\right)$ the space ${E}_{V}$ is continuously embedded in ${L}^{q}\left({ℝ}^{d}\right)$ for every $q\in \left[2,{2}^{*}\right]$, where ${2}^{*}:=\frac{2d}{d-2}$ is the classical critical Sobolev exponent. As pointed out in Introduction, one the main difficulties in working in the space ${E}_{V}$ is the lack of compactness of these embeddings.

Now, we show that an ${L}^{p}$-summability condition on the weight W allows us to consider nonlinear terms f in (Slambda) whose growth behavior is the standard ones adopted in literature studying elliptic equations defined on bounded domains.

More precisely, we have the following regularity result.

#### Proposition 2.

Assume that $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ is a continuous function such that

${\alpha }_{f}:=\underset{z\in ℝ}{sup}\frac{|f\left(z\right)|}{1+{|z|}^{q-1}}<+\mathrm{\infty },$(2.1)

where $q\mathrm{\in }\mathrm{\left[}\mathrm{2}\mathrm{,}{\mathrm{2}}^{\mathrm{*}}\mathrm{\right]}$ and let $\mathrm{\Psi }\mathrm{:}{E}_{V}\mathrm{\to }\mathrm{R}$ be defined by

where $W\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}{\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}}_{\mathrm{+}}\mathrm{\cap }{L}^{p}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}$, where $p\mathrm{:=}\frac{q}{q\mathrm{-}\mathrm{1}}$ is the conjugate Sobolev exponent of q, and $F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{:=}{\mathrm{\int }}_{\mathrm{0}}^{t}f\mathit{}\mathrm{\left(}z\mathrm{\right)}\mathit{}𝑑z$ for every $t\mathrm{\in }\mathrm{R}$. Then Ψ is continuously Gâteaux derivable on ${E}_{V}$.

#### Proof.

It is clear that Ψ is well-defined. Indeed, thanks to (2.1), our assumptions on W and the Hölder inequality ensure that

${\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left(x\right)\right)𝑑x\le {\alpha }_{f}{\int }_{{ℝ}^{d}}W\left(x\right)|u\left(x\right)|𝑑x+\frac{{\alpha }_{f}}{q}{\int }_{{ℝ}^{d}}W\left(x\right){|u\left(x\right)|}^{q}𝑑x$$\le {\alpha }_{f}{\left({\int }_{{ℝ}^{d}}{|W\left(x\right)|}^{p}𝑑x\right)}^{\frac{1}{p}}{\left({\int }_{{ℝ}^{d}}{|u\left(x\right)|}^{q}𝑑x\right)}^{\frac{1}{q}}+\frac{{\alpha }_{f}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\int }_{{ℝ}^{d}}{|u\left(x\right)|}^{q}𝑑x$(2.2)

for every $u\in {E}_{V}$. Since ${E}_{V}$ is continuously embedded in ${L}^{q}\left({ℝ}^{d}\right)$, inequality (2.2) yields

$\mathrm{\Psi }\left(u\right)\le {\alpha }_{f}\left({\parallel W\parallel }_{p}{\parallel u\parallel }_{q}+\frac{1}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\parallel u\parallel }_{q}^{q}\right)<+\mathrm{\infty }$

for every $u\in {E}_{V}$.

Now, let us compute the Gâteaux derivative ${\mathrm{\Psi }}^{\prime }:{E}_{V}\to {E}_{V}^{*}$ such that $u↦〈{\mathrm{\Psi }}^{\prime }\left(u\right),v〉$, where

$〈{\mathrm{\Psi }}^{\prime }\left(u\right),v〉:=\underset{h\to 0}{lim}\frac{\mathrm{\Psi }\left(u+hv\right)-\mathrm{\Psi }\left(u\right)}{h}=\underset{h\to 0}{lim}\frac{{\int }_{{ℝ}^{d}}W\left(x\right)\left(F\left(u\left(x\right)+hv\left(x\right)\right)-F\left(u\left(x\right)\right)\right)𝑑x}{h}.$

We claim that

$\underset{h\to {0}^{+}}{lim}\frac{{\int }_{{ℝ}^{d}}W\left(x\right)\left(F\left(u\left(x\right)+hv\left(x\right)\right)-F\left(u\left(x\right)\right)\right)𝑑x}{h}={\int }_{{ℝ}^{d}}W\left(x\right)f\left(u\left(x\right)\right)v\left(x\right)𝑑x.$(2.3)

Indeed, for a.e. $x\in {ℝ}^{d}$, and $|h|\in \left(0,1\right)$, by the Mean Value Theorem, there exists $\theta \in \left(0,1\right)$ (depending of x) such that (up to the constant ${\alpha }_{f}$)

$W\left(x\right)|f\left(u\left(x\right)+\theta hv\left(x\right)\right)||v\left(x\right)|\le W\left(x\right)\left(1+{|u\left(x\right)+\theta hv\left(x\right)|}^{q-1}\right)|v\left(x\right)|$$\le W\left(x\right)|v\left(x\right)|+{\parallel W\parallel }_{\mathrm{\infty }}{||u\left(x\right)|+|v\left(x\right)||}^{q-1}|v\left(x\right)|$$\le W\left(x\right)|v\left(x\right)|+{2}^{q-2}{\parallel W\parallel }_{\mathrm{\infty }}\left({|u\left(x\right)|}^{q-1}+{|v\left(x\right)|}^{q-1}\right)|v\left(x\right)|.$

Setting

$g\left(x\right):=W\left(x\right)|v\left(x\right)|+{2}^{q-2}{\parallel W\parallel }_{\mathrm{\infty }}\left({|u\left(x\right)|}^{q-1}+{|v\left(x\right)|}^{q-1}\right)|v\left(x\right)|$

for a.e. $x\in {ℝ}^{d}$, owing to $q\in \left[2,{2}^{*}\right]$, the Hölder inequality ensures that

${\int }_{{ℝ}^{d}}g\left(x\right)𝑑x\le {\parallel W\parallel }_{p}{\parallel v\parallel }_{q}+{2}^{q-2}{\parallel W\parallel }_{\mathrm{\infty }}{\parallel u\parallel }_{q}^{q-1}{\parallel v\parallel }_{q}+{2}^{q-2}{\parallel W\parallel }_{\mathrm{\infty }}{\parallel v\parallel }_{q}^{q}<+\mathrm{\infty }.$

Thus $g\in {L}^{1}\left({ℝ}^{d}\right)$ and Lebesgue’s Dominated Convergence Theorem ensures that (2.3) holds true.

We prove now the continuity of the Gâteaux derivative ${\mathrm{\Psi }}^{\prime }$. Assume that ${u}_{j}\to u$ in ${E}_{V}$ as $j\to +\mathrm{\infty }$, and let us prove that

$\underset{j\to +\mathrm{\infty }}{lim}{\parallel {\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\parallel }_{{E}_{V}^{*}}=0,$

where

${\parallel {\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\parallel }_{{E}_{V}^{*}}:=\underset{{\parallel v\parallel }_{{E}_{V}}=1}{sup}|〈{\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right),v〉|.$

Observe that

${\parallel {\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\parallel }_{{E}_{V}^{*}}:=\underset{{\parallel v\parallel }_{{E}_{V}}=1}{sup}|〈{\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right),v〉|$$\le \underset{{\parallel v\parallel }_{{E}_{V}}=1}{sup}|{\int }_{{ℝ}^{d}}W\left(x\right)\left(f\left({u}_{j}\left(x\right)\right)-f\left(u\left(x\right)\right)\right)v\left(x\right)𝑑x|$$\le \underset{{\parallel v\parallel }_{{E}_{V}}=1}{sup}{\int }_{{ℝ}^{d}}W\left(x\right)|\left(f\left({u}_{j}\left(x\right)\right)-f\left(u\left(x\right)\right)\right)||v\left(x\right)|𝑑x.$(2.4)

The Sobolev imbedding result ensures that ${u}_{j}\to u$ in ${L}^{q}\left({ℝ}^{n}\right)$ for every $q\in \left[2,{2}^{*}\right]$, and ${u}_{j}\left(x\right)\to u\left(x\right)$ a.e. in ${ℝ}^{d}$ as $j\to +\mathrm{\infty }$. Then, by [42, Lemma A.1], up to a subsequence, ${u}_{j}\left(x\right)\to u\left(x\right)$ a.e. in ${ℝ}^{d}$ as $j\to +\mathrm{\infty }$, and there exists $\tau \in {L}^{q}\left({ℝ}^{d}\right)$ such that $|{u}_{j}\left(x\right)|\le \tau \left(x\right)$ a.e. in ${ℝ}^{d}$ and $|u\left(x\right)|\le \tau \left(x\right)$ a.e. in ${ℝ}^{d}$. In particular,

Note that the function ${ℝ}^{d}\ni x↦1+\tau {\left(x\right)}^{q-1}$ is $p:=\frac{q}{q-1}$-summable on every bounded measurable subset of ${ℝ}^{d}$. Therefore, by the Lebesgue Dominated Convergence Theorem, we infer

for every $R>0$, where ${B}_{R}\left(0\right)$ denotes the open ball of radius R centered at zero. At this point, set

${\mathrm{\ell }}_{q}:=\underset{{\parallel v\parallel }_{{E}_{V}}\le 1}{sup}{\parallel v\parallel }_{q}$

and let ${R}_{\epsilon }>0$ be such that

${\int }_{{ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)}{|W\left(x\right)|}^{p}𝑑x<{\left(\frac{\epsilon }{8{\alpha }_{f}{\mathrm{\ell }}_{q}}\right)}^{p},$(2.5)

as well as

${\int }_{{ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)}{|\tau \left(x\right)|}^{p}𝑑x<{\left(\frac{\epsilon }{8{\alpha }_{f}{\mathrm{\ell }}_{q}{\parallel W\parallel }_{\mathrm{\infty }}}\right)}^{p}.$(2.6)

Moreover, let ${j}_{\epsilon }\in ℕ$ be such that

${\parallel f\left({u}_{j}\right)-f\left(u\right)\parallel }_{{L}^{p}\left({B}_{{R}_{\epsilon }}\left(0\right)\right)}<\frac{\epsilon }{4{\mathrm{\ell }}_{q}{\parallel W\parallel }_{\mathrm{\infty }}}$(2.7)

for each $j\in ℕ$, with $j\ge {j}_{\epsilon }$. Owing to (2.5), (2.6) and (2.7), for $j\ge {j}_{\epsilon }$, inequality (2.4) yields

${\parallel {\mathrm{\Psi }}^{\prime }\left({u}_{j}\right)-{\mathrm{\Psi }}^{\prime }\left(u\right)\parallel }_{{E}_{V}^{*}}\le \underset{{\parallel v\parallel }_{{E}_{V}}\le 1}{sup}{\int }_{{ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)}W\left(x\right)|f\left({u}_{j}\left(x\right)\right)-f\left(u\left(x\right)\right)||v\left(x\right)|𝑑x$$\le 2{\alpha }_{f}\underset{{\parallel v\parallel }_{{E}_{V}}\le 1}{sup}{\int }_{{ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)}W\left(x\right)\left(1+{|\tau \left(x\right)|}^{q-1}\right)|v\left(x\right)|𝑑x$$+{\parallel W\parallel }_{\mathrm{\infty }}\underset{{\parallel v\parallel }_{{E}_{V}}\le 1}{sup}{\int }_{{B}_{{R}_{\epsilon }}\left(0\right)}|f\left({u}_{j}\left(x\right)\right)-f\left(u\left(x\right)\right)||v\left(x\right)|𝑑x$$\le 2{\alpha }_{f}{\mathrm{\ell }}_{q}{\parallel W\parallel }_{{L}^{p}\left({ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)\right)}+2{\alpha }_{f}{\mathrm{\ell }}_{q}{\parallel W\parallel }_{\mathrm{\infty }}{\parallel \tau \parallel }_{{L}^{p}\left({ℝ}^{d}\setminus {B}_{{R}_{\epsilon }}\left(0\right)\right)}$$+{\mathrm{\ell }}_{q}{\parallel W\parallel }_{\mathrm{\infty }}{\parallel f\left({u}_{j}\right)-f\left(u\right)\parallel }_{{L}^{p}\left({B}_{{R}_{\epsilon }}\left(0\right)\right)}<\epsilon ,$

that concludes the proof. ∎

Of course, by elementary standard arguments, Proposition 2 ensures that $\mathrm{\Psi }\in {C}^{1}\left({E}_{V},ℝ\right)$. Then the functional defined by

is of class ${C}^{1}\left({E}_{V},ℝ\right)$. Furthermore, the critical points of ${J}_{\lambda }$ are exactly the weak solutions of problem (Slambda). More precisely, we say that $u\in {E}_{V}$ is a weak solution of problem (Slambda) if and only if

In order to find critical points for ${J}_{\lambda }$, we will apply the principle of symmetric criticality together with the following critical point theorem proved by Ricceri in [38] and recalled here in a more convenient form.

#### Theorem 3.

Let X be a reflexive real Banach space, and let $\mathrm{\Phi }\mathrm{,}\mathrm{\Psi }\mathrm{:}X\mathrm{\to }\mathrm{R}$ be two Gâteaux differentiable functionals such that Φ is strongly continuous, sequentially weakly lower semicontinuous and coercive. Further, assume that Ψ is sequentially weakly upper semicontinuous. For every $r\mathrm{>}{\mathrm{inf}}_{X}\mathit{}\mathrm{\Phi }$, put

$\phi \left(r\right):=\underset{u\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },r\right)\right)}{inf}\frac{\left({sup}_{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },r\right)\right)}\mathrm{\Psi }\left(v\right)\right)-\mathrm{\Psi }\left(u\right)}{r-\mathrm{\Phi }\left(u\right)}.$

Then, for each $r\mathrm{>}{\mathrm{inf}}_{X}\mathit{}\mathrm{\Phi }$ and each $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\frac{\mathrm{1}}{\phi \mathit{}\mathrm{\left(}r\mathrm{\right)}}\mathrm{\right)}$, the restriction of ${J}_{\lambda }\mathrm{:=}\mathrm{\Phi }\mathrm{-}\lambda \mathit{}\mathrm{\Psi }$ to ${\mathrm{\Phi }}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\left(}\mathrm{-}\mathrm{\infty }\mathrm{,}r\mathrm{\right)}\mathrm{\right)}$ admits a global minimum, which is a critical point $\mathrm{\left(}$local minimum$\mathrm{\right)}$ of ${J}_{\lambda }$ in X.

For completeness, we also recall here the principle of symmetric criticality that plays a central role in many problems from the differential geometry, physics and in partial differential equations.

An action of a topological group G on the Banach space $\left(X,\parallel \cdot {\parallel }_{X}\right)$ is a continuous map

$*:G×X\to X,\left(g,x\right)↦g*u$

such that

The action $*$ is said to be isometric if ${\parallel g*u\parallel }_{X}={\parallel u\parallel }_{X}$ for every $g\in G$. Moreover, the space of G-invariant points is defined by

and a map $h:X\to ℝ$ is said to be G-invariant on X if

$h\left(g*u\right)=h\left(u\right)$

for every $g\in G$ and $u\in X$.

#### Theorem 4 (Palais (1979), [36]).

Assume that the action of the topological group G on the Banach space X is isometric. If $I\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}X\mathrm{,}\mathrm{R}\mathrm{\right)}$ is G-invariant on X and if u is a critical point of I restricted to ${\mathrm{Fix}}_{G}\mathit{}\mathrm{\left(}X\mathrm{\right)}$, then u is a critical point of I.

See, for instance, [42, Chapter 1] for details.

Let $O\left(d\right)$ be the orthogonal group and let $G\subseteq O\left(d\right)$ be a subgroup. Assume that G acts on the space ${E}_{V}$. Hence, the set of fixed points of ${E}_{V}$, respect to $O\left(d\right)$, is clearly given by

We notice that, if $G=O\left(d\right)$ and the action is the standard linear isometric map defined by

then ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ is exactly the subspace of radially symmetric functions of ${E}_{V}$. Moreover, the embedding

${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)↪{L}^{q}\left({ℝ}^{d}\right)$(2.8)

is continuous (respectively, compact) for every $q\in \left[2,{2}^{*}\right]$ (respectively, $q\in \left(2,{2}^{*}\right)$). See, for instance, the celebrated paper [31].

Now, for every $\lambda >0,$ let ${\mathcal{𝒥}}_{\lambda }:={{J}_{\lambda }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}:{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)\to ℝ$ be the functional defined by

${\mathcal{𝒥}}_{\lambda }\left(u\right):=\mathrm{\Phi }\left(u\right)-{\lambda \mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right),$

where

The above remarks yield the next semicontinuity property.

#### Lemma 5.

Assume that $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ is a continuous function such that condition (2.1) holds for every $q\mathrm{\in }\mathrm{\left(}\mathrm{2}\mathrm{,}{\mathrm{2}}^{\mathrm{*}}\mathrm{\right)}$. Then, for every $\lambda \mathrm{>}\mathrm{0}$, the functional ${\mathcal{J}}_{\lambda }$ is sequentially weakly lower semicontinuous on ${\mathrm{Fix}}_{O\mathit{}\mathrm{\left(}d\mathrm{\right)}}\mathit{}\mathrm{\left(}{E}_{V}\mathrm{\right)}$.

#### Proof.

First, on account of Brézis [16, Corollaire III.8], the functional Φ is sequentially weakly lower semicontinuous on ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$. In order to prove that ${\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}$ is sequentially weakly continuous, we assume that there exists a sequence ${\left\{{u}_{j}\right\}}_{j\in ℕ}\subset {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ which weakly converges to an element ${u}_{0}\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$. Since ${\left\{{u}_{j}\right\}}_{j\in ℕ}$ is bounded in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ and, taking into account that, thanks to (2.8), ${u}_{j}\to {u}_{0}$ in ${L}^{q}\left({ℝ}^{d}\right)$, the Mean Value Theorem, the growth condition (2.1) and the Hölder inequality yield

$|\mathrm{\Psi }\left({u}_{j}\right)-\mathrm{\Psi }\left({u}_{0}\right)|\le {\int }_{{ℝ}^{d}}W\left(x\right)|F\left({u}_{j}\left(x\right)\right)-F\left({u}_{0}\left(x\right)\right)|𝑑x$$\le {\alpha }_{f}{\int }_{{ℝ}^{d}}W\left(x\right)\left(2+{|{u}_{j}\left(x\right)|}^{q-1}+{|{u}_{0}\left(x\right)|}^{q-1}\right)|{u}_{j}\left(x\right)-{u}_{0}\left(x\right)|𝑑x$$\le {\alpha }_{f}\left(2{\parallel W\parallel }_{p}{\parallel {u}_{j}-{u}_{0}\parallel }_{q}+{\parallel W\parallel }_{\mathrm{\infty }}\left({\parallel {u}_{j}\parallel }_{q}^{q-1}+{\parallel {u}_{0}\parallel }_{q}^{q-1}\right){\parallel {u}_{j}-{u}_{0}\parallel }_{q}\right)$$\le {\alpha }_{f}\left(2{\parallel W\parallel }_{2}+{\parallel W\parallel }_{\mathrm{\infty }}\left(M+{\parallel {u}_{0}\parallel }_{q}^{q-1}\right)\right){\parallel {u}_{j}-{u}_{0}\parallel }_{q}$(2.9)

for some $M>0$. The last expression in (2.9) tends to zero. In conclusion, the functional Ψ is sequentially weakly continuous and this completes the proof. ∎

## 3 An existence result: A local minimum approach

Let $d\ge 3$ and set

${c}_{\mathrm{\ell }}:=sup\left\{\frac{{\parallel u\parallel }_{\mathrm{\ell }}}{{\parallel u\parallel }_{{E}_{V}}}:u\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)\setminus \left\{0\right\}\right\}$

for every $\mathrm{\ell }\in \left(2,{2}^{*}\right)$.

With the above notation the main result reads as follows.

#### Theorem 6.

Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be a continuous function satisfying the growth condition (2.1) for some $q\mathrm{\in }\mathrm{\left(}\mathrm{2}\mathrm{,}{\mathrm{2}}^{\mathrm{*}}\mathrm{\right)}$ in addition to

$\underset{t\to {0}^{+}}{lim sup}\frac{F\left(t\right)}{{t}^{2}}=+\mathrm{\infty }\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\underset{t\to {0}^{+}}{lim inf}\frac{F\left(t\right)}{{t}^{2}}>-\mathrm{\infty },$(3.1)

where $F\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{:=}{\mathrm{\int }}_{\mathrm{0}}^{t}f\mathit{}\mathrm{\left(}z\mathrm{\right)}\mathit{}𝑑z$. Furthermore, let $W\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\cap }{L}^{p}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$, be a radially symmetric map with

and V be a potential for which $\mathrm{\left(}{h}_{V}^{d}\mathrm{\right)}$ holds. Then there exists a positive number ${\lambda }^{\mathrm{\star }}$ given by

${\lambda }^{\star }:=\frac{q}{{\alpha }_{f}{c}_{q}}\underset{\gamma >0}{\mathrm{max}}\left(\frac{\gamma }{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\gamma }^{q-1}}\right)$(3.2)

such that, for every $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\lambda }^{\mathrm{\star }}\mathrm{\right)}$, the problem

(Slambda)

admits at least one non-trivial radial weak solution ${u}_{\lambda }\mathrm{\in }{E}_{V}$. Moreover,

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=0,$

and $\mathrm{|}{u}_{\lambda }\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{|}\mathrm{\to }\mathrm{0}$ as $\mathrm{|}x\mathrm{|}\mathrm{\to }\mathrm{\infty }$.

#### Proof.

The main idea of the proof consists in applying Theorem 3 to the functional ${\mathcal{𝒥}}_{\lambda }$. Successively, taking into account the preliminary results of Section 2, the existence of one non-trivial radial solution of problem (Slambda) follows by the symmetric criticality principle recalled in Theorem 4. To this purpose, we write the functional ${\mathcal{𝒥}}_{\lambda }$ as follows:

with

$\mathrm{\Phi }\left(u\right):=\frac{1}{2}\left({\int }_{{ℝ}^{d}}{|\nabla u\left(x\right)|}^{2}𝑑x+{\int }_{{ℝ}^{d}}V\left(x\right){|u\left(x\right)|}^{2}𝑑x\right)$

as well as

$\mathrm{\Psi }\left(u\right):={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left(x\right)\right)𝑑x.$

First of all, note that ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ is a Hilbert space and the functionals Φ and ${\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}$ have the regularity required by Theorem 3 (see Lemma 5). Moreover, it is clear that the functional Φ is strongly continuous, coercive in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ and

$\underset{u\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}{inf}\mathrm{\Phi }\left(u\right)=0.$

Now, since $0<\lambda <{\lambda }^{\star }$, bearing in mind (3.2), there exists $\overline{\gamma }>0$ such that

$\lambda <{\lambda }^{\star }\left(\overline{\gamma }\right):=\frac{q}{{\alpha }_{f}{c}_{q}}\left(\frac{\overline{\gamma }}{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-1}}\right).$(3.3)

Set $r\in \left(0,+\mathrm{\infty }\right)$ and consider the function $\chi :\left(0,+\mathrm{\infty }\right)\to \left[0,+\mathrm{\infty }\right)$ given by

$\chi \left(r\right):=\frac{{{sup}_{u\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },r\right)\right)}\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)}{r}.$

By taking into account the growth condition expressed by (2.1), it follows that

${\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left(x\right)\right)𝑑x\le {\alpha }_{f}{\int }_{{ℝ}^{d}}W\left(x\right)|u\left(x\right)|𝑑x+\frac{{\alpha }_{f}}{q}{\int }_{{ℝ}^{d}}W\left(x\right){|u\left(x\right)|}^{q}𝑑x.$

Moreover, one has

${\parallel u\parallel }_{{E}_{V}}<\sqrt{2r}$(3.4)

for every $u\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ and $\mathrm{\Phi }\left(u\right).

Now, by using (3.4), the Sobolev embedding (2.8) yields

${\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)<{\alpha }_{f}{c}_{q}\left({\parallel W\parallel }_{p}\sqrt{2r}+\frac{{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\left(2r\right)}^{\frac{q}{2}}\right),$

for every $u\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ such that $\mathrm{\Phi }\left(u\right). Hence

${\underset{u\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },r\right)\right)}{sup}\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)\le {\alpha }_{f}{c}_{q}\left({\parallel W\parallel }_{p}\sqrt{2r}+\frac{{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\left(2r\right)}^{\frac{q}{2}}\right).$

Then the above inequality immediately gives

$\chi \left(r\right)\le {\alpha }_{f}{c}_{q}\left({\parallel W\parallel }_{p}\sqrt{\frac{2}{r}}+\frac{{2}^{\frac{q}{2}}{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{r}^{\frac{q}{2}-1}\right)$(3.5)

for every $r>0$.

Evaluating inequality (3.5) in $r={\overline{\gamma }}^{2}$, we have

$\chi \left({\overline{\gamma }}^{2}\right)\le {\alpha }_{f}{c}_{q}\left(\sqrt{2}\frac{{\parallel W\parallel }_{p}}{\overline{\gamma }}+\frac{{2}^{\frac{q}{2}}{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-2}\right).$(3.6)

Now, it is easy to note that

$\phi \left({\overline{\gamma }}^{2}\right):=\underset{u\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\frac{\left({{sup}_{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)}{r-\mathrm{\Phi }\left(u\right)}\le \chi \left({\overline{\gamma }}^{2}\right),$

owing to ${z}_{0}\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)$ and $\mathrm{\Phi }\left({z}_{0}\right)={\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({z}_{0}\right)=0$, where ${z}_{0}\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ is the zero function.

Finally, bearing in mind (3.3), the above inequality together with (3.6) produce

$\phi \left({\overline{\gamma }}^{2}\right)\le \chi \left({\overline{\gamma }}^{2}\right)\le {\alpha }_{f}{c}_{q}\left(\sqrt{2}\frac{{\parallel W\parallel }_{p}}{\overline{\gamma }}+\frac{{2}^{\frac{q}{2}}{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-2}\right)<\frac{1}{\lambda }.$(3.7)

Hence, we have that

$\lambda \in \left(0,\frac{q}{{\alpha }_{f}{c}_{q}}\left(\frac{\overline{\gamma }}{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-1}}\right)\right)\subseteq \left(0,\frac{1}{\phi \left({\overline{\gamma }}^{2}\right)}\right).$

The (critical point) Theorem 3 ensures that there exists a function ${u}_{\lambda }\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)$ such that

${\mathrm{\Phi }}^{\prime }\left({u}_{\lambda }\right)-\lambda {\left({\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\right)}^{\prime }\left({u}_{\lambda }\right)=0$

and, in particular, ${u}_{\lambda }$ is a global minimum of the restriction of the functional ${\mathcal{𝒥}}_{\lambda }$ to the sublevel ${\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)$.

Now, we have to show that the solution ${u}_{\lambda }$ found here above is not the trivial (identically zero) function. If $f\left(0\right)\ne 0$, then it easily follows that ${u}_{\lambda }\not\equiv 0$ in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$, since the trivial function does not solve problem (Slambda). Let us consider the case when $f\left(0\right)=0$ and let us fix $\lambda \in \left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$ for some $\overline{\gamma }>0$. Finally, let ${u}_{\lambda }$ be such that

(3.8)

and

$\mathrm{\Phi }\left({u}_{\lambda }\right)<{\overline{\gamma }}^{2},$

and also ${u}_{\lambda }$ is a critical point of ${\mathcal{𝒥}}_{\lambda }$ in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$. Since $O\left(d\right)$ acts isometrically on ${E}_{V}$ (note that V is radial) and taking into account that, thanks to the symmetry assumption on W, one has

${\int }_{{ℝ}^{d}}W\left(x\right)F\left(\left(gu\right)\left(x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left({g}^{-1}x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(y\right)F\left(u\left(y\right)\right)𝑑y$

for every $g\in O\left(d\right)$, the functional ${J}_{\lambda }$ is $O\left(d\right)$-invariant on ${E}_{V}$.

So, owing to Theorem 4, ${u}_{\lambda }$ is a weak solution of problem (Slambda). In this setting, in order to prove that ${u}_{\lambda }\not\equiv 0$ in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$, first we claim that there exists a sequence of functions ${\left\{{w}_{j}\right\}}_{j\in ℕ}$ in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ such that

$\underset{j\to +\mathrm{\infty }}{lim sup}\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}=+\mathrm{\infty }.$(3.9)

By the assumption on the limsup in (3.1) there exists a sequence ${\left\{{t}_{j}\right\}}_{j\in ℕ}\subset \left(0,+\mathrm{\infty }\right)$ such that ${t}_{j}\to {0}^{+}$ as $j\to +\mathrm{\infty }$ and

$\underset{j\to +\mathrm{\infty }}{lim}\frac{F\left({t}_{j}\right)}{{t}_{j}^{2}}=+\mathrm{\infty },$

namely, we have that for any $M>0$ and j sufficiently large,

$F\left({t}_{j}\right)>M{t}_{j}^{2}.$(3.10)

Now, following Kristály, Moroşanu, and O’Regan in [29], we construct a special test function belonging to ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ that will be useful for our purposes. If $a, define

${A}_{a}^{b}:=\left\{x\in {ℝ}^{d}:a\le |x|\le b\right\}.$

Since $W\in {L}^{\mathrm{\infty }}\left({ℝ}^{d}\right)\setminus \left\{0\right\}$ is a radially symmetric function with $W\ge 0$, one can find two real numbers $R>r>0$ and $\alpha >0$ such that

${\mathrm{essinf}}_{x\in {A}_{r}^{R}}W\left(x\right)\ge \alpha >0.$(3.11)

Hence, let $0 such that (3.11) holds and $\sigma \in \left(0,\frac{1}{2}\left(R-r\right)\right)$. Set ${v}_{\sigma }\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ given by

where ${z}_{+}:=\mathrm{max}\left\{0,z\right\}$. With the above notation, we have:

• (1)

$\text{supp}\left({v}_{\sigma }\right)\subseteq {A}_{r}^{R}$,

• (2)

${\parallel {v}_{\sigma }\parallel }_{\mathrm{\infty }}\le 1$,

• (3)

${v}_{\sigma }\left(x\right)=1$ for every $x\in {A}_{r+\sigma }^{R-\sigma }$.

Now, define ${w}_{j}:={t}_{j}{v}_{\sigma }$ for any $j\in ℕ$. Since ${v}_{\sigma }\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$, of course one has ${w}_{j}\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ for any $j\in ℕ$. Furthermore, taking into account the algebraic properties of the functions ${v}_{\sigma }$ stated in 13, since $F\left(0\right)=0$, and by using (3.10) we can write

$\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}=\frac{{\int }_{{A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({w}_{j}\left(x\right)\right)𝑑x+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({w}_{j}\left(x\right)\right)𝑑x}{\mathrm{\Phi }\left({w}_{j}\right)}$$=\frac{{\int }_{{A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({t}_{j}\right)𝑑x+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({t}_{j}{v}_{\sigma }\left(x\right)\right)𝑑x}{\mathrm{\Phi }\left({w}_{j}\right)}$$\ge 2\frac{M|{A}_{r+\sigma }^{R-\sigma }|\alpha {t}_{j}^{2}+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({t}_{j}{v}_{\sigma }\left(x\right)\right)𝑑x}{{t}_{j}^{2}{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$(3.12)

for j sufficiently large. Now, we have to distinguish two different cases, i.e. the case when the liminf in (3.1) is $+\mathrm{\infty }$ and the one in which the liminf in (3.1) is finite.

Case 1. Suppose that ${lim}_{t\to {0}^{+}}\frac{F\left(t\right)}{{t}^{2}}=+\mathrm{\infty }$. Then there exists ${\rho }_{M}>0$ such that for any t with $0,

$F\left(t\right)\ge M{t}^{2}.$(3.13)

Since ${t}_{j}\to {0}^{+}$ and $0\le {v}_{\sigma }\left(x\right)\le 1$ in ${ℝ}^{d}$, it follows that ${w}_{j}\left(x\right)={t}_{j}{v}_{\sigma }\left(x\right)\to {0}^{+}$ as $j\to +\mathrm{\infty }$ uniformly in $x\in {ℝ}^{d}$. Hence, $0\le {w}_{j}\left(x\right)<{\rho }_{M}$ for j sufficiently large and for any $x\in {ℝ}^{d}$. Hence, as a consequence of (3.12) and (3.13), we have

$\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}\ge 2\frac{M|{A}_{r+\sigma }^{R-\sigma }|\alpha {t}_{j}^{2}+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({t}_{j}{v}_{\sigma }\left(x\right)\right)𝑑x}{{t}_{j}^{2}{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$$\ge 2M\alpha \frac{|{A}_{r+\sigma }^{R-\sigma }|+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x}{{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$

for j sufficiently large. The arbitrariness of M gives (3.9) and so the claim is proved.

Case 2. Suppose that ${lim inf}_{t\to {0}^{+}}\frac{F\left(t\right)}{{t}^{2}}=\mathrm{\ell }\in ℝ$. Then, for any $\epsilon >0$, there exists ${\rho }_{\epsilon }>0$ such that for any t with $0,

$F\left(t\right)\ge \left(\mathrm{\ell }-\epsilon \right){t}^{2}.$(3.14)

Arguing as above, we can suppose that $0\le {w}_{j}\left(x\right)={t}_{j}{v}_{\sigma }\left(x\right)<{\rho }_{\epsilon }$ for j large enough and any $x\in {ℝ}^{d}$. Thus, by (3.12) and (3.14) we get

$\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}\ge 2\frac{M|{A}_{r+\sigma }^{R-\sigma }|\alpha {t}_{j}^{2}+{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}W\left(x\right)F\left({t}_{j}{v}_{\sigma }\left(x\right)\right)𝑑x}{{t}_{j}^{2}{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$$\ge 2\alpha \frac{M|{A}_{r+\sigma }^{R-\sigma }|+\left(\mathrm{\ell }-\epsilon \right){\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x}{{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}},$(3.15)

provided j is sufficiently large. Choosing $M>0$ large enough, say

$M>\mathrm{max}\left\{0,-\frac{2\mathrm{\ell }}{|{A}_{r+\sigma }^{R-\sigma }|}{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x\right\},$

and $\epsilon >0$ small enough so that

$\epsilon {\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x<\frac{M}{2}|{A}_{r+\sigma }^{R-\sigma }|+\mathrm{\ell }{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x,$

by (3.15) we get

$\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}\ge 2\alpha \frac{M|{A}_{r+\sigma }^{R-\sigma }|+\left(\mathrm{\ell }-\epsilon \right){\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x}{{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$$\ge \frac{2\alpha }{{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}\left(M|{A}_{r+\sigma }^{R-\sigma }|+\mathrm{\ell }{\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x-\epsilon {\int }_{{A}_{r}^{R}\setminus {A}_{r+\sigma }^{R-\sigma }}{|{v}_{\sigma }\left(x\right)|}^{2}𝑑x\right)$$=\alpha M\frac{|{A}_{r+\sigma }^{R-\sigma }|}{{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}^{2}}$

for j large enough. Also in this case the arbitrariness of M gives assertion (3.9). Now, note that

${\parallel {w}_{j}\parallel }_{{E}_{V}}={t}_{j}{\parallel {v}_{\sigma }\parallel }_{{E}_{V}}\to 0$

as $j\to +\mathrm{\infty }$, so that for j large enough,

${\parallel {w}_{j}\parallel }_{{E}_{V}}<\sqrt{2}\overline{\gamma }.$

Thus

${w}_{j}\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right),$(3.16)

provided j is large enough. Also, by ((3.9)) and the fact that $\lambda >0$

${\mathcal{𝒥}}_{\lambda }\left({w}_{j}\right)=\mathrm{\Phi }\left({w}_{j}\right)-{\lambda \mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)<0$(3.17)

for j sufficiently large. Since ${u}_{\lambda }$ is a global minimum of the restriction of ${\mathcal{𝒥}}_{\lambda }$ to ${\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)$ (see (3.8)), by (3.16) and (3.17) we conclude that

${\mathcal{𝒥}}_{\lambda }\left({u}_{\lambda }\right)\le {\mathcal{𝒥}}_{\lambda }\left({w}_{j}\right)<0={\mathcal{𝒥}}_{\lambda }\left(0\right),$(3.18)

so that ${u}_{\lambda }\not\equiv 0$ in ${\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$. Thus, ${u}_{\lambda }$ is a non-trivial weak solution of problem (Slambda). The arbitrariness of λ gives that ${u}_{\lambda }\not\equiv 0$ for any $\lambda \in \left(0,{\lambda }^{\star }\right)$.

Now, we claim that ${lim}_{\lambda \to {0}^{+}}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=0.$ For this, let us fix $\lambda \in \left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$ for some $\overline{\gamma }>0$. By $\mathrm{\Phi }\left({u}_{\lambda }\right)<{\overline{\gamma }}^{2}$ one has

$\mathrm{\Phi }\left({u}_{\lambda }\right)=\frac{1}{2}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}^{2}<{\overline{\gamma }}^{2},$

that is,

${\parallel {u}_{\lambda }\parallel }_{{E}_{V}}<\sqrt{2}\overline{\gamma }.$

As a consequence of this and by using the growth condition (2.1) together with the property (2.8), it follows that

$|{\int }_{{ℝ}^{d}}W\left(x\right)f\left({u}_{\lambda }\left(x\right)\right){u}_{\lambda }\left(x\right)𝑑x|\le {\alpha }_{f}\left({\int }_{{ℝ}^{d}}W\left(x\right)|{u}_{\lambda }\left(x\right)|𝑑x+{\int }_{{ℝ}^{d}}W\left(x\right){|{u}_{\lambda }\left(x\right)|}^{q}𝑑x\right)$$\le {\alpha }_{f}\left({\parallel W\parallel }_{p}{\parallel {u}_{\lambda }\parallel }_{q}+{\parallel W\parallel }_{\mathrm{\infty }}{\parallel {u}_{\lambda }\parallel }_{q}^{q}\right)$$<{c}_{q}{\alpha }_{f}\left(\sqrt{2}\parallel W{\parallel }_{p}\overline{\gamma }+{2}^{\frac{q}{2}}{c}_{q}^{q-1}\parallel W{\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q}\right)=:{M}_{\overline{\gamma }}.$(3.19)

Since ${u}_{\lambda }$ is a critical point of ${\mathcal{𝒥}}_{\lambda }$, it follows that $〈{\mathcal{𝒥}}_{\lambda }^{\prime }\left({u}_{\lambda }\right),\phi 〉=0$ for any $\phi \in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ and every $\lambda \in \left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$. In particular, $〈{\mathcal{𝒥}}_{\lambda }^{\prime }\left({u}_{\lambda }\right),{u}_{\lambda }〉=0$, that is,

$〈{\mathrm{\Phi }}^{\prime }\left({u}_{\lambda }\right),{u}_{\lambda }〉=\lambda {\int }_{{ℝ}^{d}}W\left(x\right)f\left({u}_{\lambda }\left(x\right)\right){u}_{\lambda }\left(x\right)𝑑x$(3.20)

for every $\lambda \in \left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$. Then, from (3.19) and (3.20), it follows that

$0\le {\parallel {u}_{\lambda }\parallel }_{{E}_{V}}^{2}=〈{\mathrm{\Phi }}^{\prime }\left({u}_{\lambda }\right),{u}_{\lambda }〉=\lambda {\int }_{{ℝ}^{d}}W\left(x\right)f\left({u}_{\lambda }\left(x\right)\right){u}_{\lambda }\left(x\right)𝑑x<\lambda {M}_{\overline{\gamma }}$

for any $\lambda \in \left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$. We get ${lim}_{\lambda \to {0}^{+}}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=0$, as claimed.

Finally, a Strauss-type estimate (see [31] for details) proves that the functions $u\in {E}_{V}$ are homoclinic. Hence, the solution ${u}_{\lambda }\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)\setminus \left\{0\right\}\subseteq {E}_{V}$ has this property. This concludes the proof of Theorem 6. ∎

#### Remark 7.

Theorem 6 has been achieved without any use of the well-known Ambrosetti–Rabinowitz condition. The importance of this assumption is due to the fact that it assures the boundedness of the Palais–Smale sequences for the energy functional associated with the problem under consideration. This condition fails when dealing with some superlinear elliptic boundary value problems (see, for instance, [39] and the references therein).

#### Remark 8.

If the nonlinear term f has the asymptotic behavior

$\underset{t\to {0}^{+}}{lim}\frac{f\left(t\right)}{t}=+\mathrm{\infty },$

then, obviously, hypothesis (3.1) in Theorem 6 is verified. Actually, the condition

$\underset{t\to {0}^{+}}{lim inf}\frac{F\left(t\right)}{{t}^{2}}>-\mathrm{\infty }$

is a technicality that we request in our proof in order to show that there exists a sequence ${\left\{{w}_{j}\right\}}_{j\in ℕ}\subset {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$ such that

$\underset{j\to +\mathrm{\infty }}{lim sup}\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({w}_{j}\right)}{\mathrm{\Phi }\left({w}_{j}\right)}=+\mathrm{\infty }.$

It is natural to ask if this assumption can be dropped in Theorem 6 requiring only

$\underset{t\to {0}^{+}}{lim sup}\frac{F\left(t\right)}{{t}^{2}}=+\mathrm{\infty }.$

A careful analysis of the proof of Theorem 6 ensures that the result remains valid for the following problem:

where $f:{ℝ}^{d}×ℝ\to ℝ$ is a continuous function that satisfies:

• (f1)

There exists $q\mathrm{\in }\mathrm{\left(}\mathrm{2}\mathrm{,}{\mathrm{2}}^{\mathrm{*}}\mathrm{\right)}$ such that

for some radially symmetric function $W\mathrm{\in }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\cap }{L}^{\frac{q}{q\mathrm{-}\mathrm{1}}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{d}\mathrm{\right)}\mathrm{\setminus }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$ .

• (f2)

For every $x\mathrm{\in }{\mathrm{R}}^{d}$ and $t\mathrm{\in }\mathrm{R}$ it follows that

$F\left(x,t\right)=F\left(gx,t\right)$

for every $g\mathrm{\in }O\mathit{}\mathrm{\left(}d\mathrm{\right)}$ .

• (f3)

For some ${x}_{\mathrm{0}}\mathrm{\in }{\mathrm{R}}^{d}$, there exists ${\varrho }_{\mathrm{0}}\mathrm{>}\mathrm{0}$ such that

$\underset{t\to {0}^{+}}{lim sup}\frac{{inf}_{B\left({x}_{0},{\varrho }_{0}\right)}F\left(x,t\right)}{{t}^{2}}=+\mathrm{\infty }\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\underset{t\to {0}^{+}}{lim inf}\frac{{inf}_{B\left({x}_{0},{\varrho }_{0}\right)}F\left(x,t\right)}{{t}^{2}}>-\mathrm{\infty }.$

Here $F\left(x,t\right):={\int }_{0}^{t}f\left(x,z\right)𝑑z$ and $B\left({x}_{0},{\varrho }_{0}\right)$ is the closed ball centered in ${x}_{0}$ and radius ${\varrho }_{0}$. See [27, 26] for related topics.

#### Remark 9.

We perform now the behavior of the functional ${\mathcal{𝒥}}_{\lambda }$ depending on the real parameter λ. In particular, we point out that by (3.18) the map

(3.21)

Moreover, fixing $\overline{\gamma }>0$, the function $\lambda ↦{\mathcal{𝒥}}_{\lambda }\left({u}_{\lambda }\right)$ is strictly decreasing in $\left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$, where

${\lambda }^{\star }\left(\overline{\gamma }\right):=\frac{q}{{\alpha }_{f}{c}_{q}}\left(\frac{\overline{\gamma }}{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-1}}\right).$

Indeed, let us write

${\mathcal{𝒥}}_{\lambda }\left(u\right)=\lambda \left(\frac{\mathrm{\Phi }\left(u\right)}{\lambda }-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(u\right)\right)$(3.22)

for every $u\in {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)$. Now, fix $0<{\lambda }_{1}<{\lambda }_{2}<{\lambda }^{\star }\left(\overline{\gamma }\right)$ and let ${u}_{{\lambda }_{i}}$ be the global minimum of the functional ${\mathcal{𝒥}}_{{\lambda }_{i}}$ restricted to the sublevel ${\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)$ for $i=1,2$. We observe that

$\left(\frac{\mathrm{\Phi }\left({u}_{{\lambda }_{i}}\right)}{{\lambda }_{i}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left({u}_{{\lambda }_{i}}\right)\right)=\underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{i}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)$

for every $i=1,2$. Of course, (3.21) and (3.22) yield

Moreover, it is easy to note that

$\underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{2}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)\le \underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{1}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)$(3.23)

owing to $0<{\lambda }_{1}<{\lambda }_{2}$. Then, by (3.22)–(3.23) and again by the fact that $0<{\lambda }_{1}<{\lambda }_{2}$, we have

${\mathcal{𝒥}}_{{\lambda }_{2}}\left({u}_{{\lambda }_{2}}\right)={\lambda }_{2}\underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },\overline{\gamma }{}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{2}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)$$\le {\lambda }_{2}\underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{1}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)$$<{\lambda }_{1}\underset{v\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)}{inf}\left(\frac{\mathrm{\Phi }\left(v\right)}{{\lambda }_{1}}-{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)}\left(v\right)\right)$$={J}_{{\lambda }_{1}}\left({u}_{{\lambda }_{1}}\right),$

so that the real function $\lambda ↦{\mathcal{𝒥}}_{\lambda }\left({u}_{\lambda }\right)$ is (strictly) decreasing in $\left(0,{\lambda }^{\star }\left(\overline{\gamma }\right)\right)$.

## 4 Multiple solutions in presence of symmetries

We study now the existence of multiple solutions (radial and non-radial) for Schrödinger equations in presence of a symmetric nonlinear term f.

Let either $d=4$ or $d\ge 6$ and consider the subgroup ${H}_{d,i}\subset O\left(d\right)$ given by

for every $i\in J:=\left\{1,\mathrm{\dots },{\tau }_{d}\right\}$, where

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right].$

Let us define the involution ${\eta }_{{H}_{d,i}}:{ℝ}^{d}\to {ℝ}^{d}$ as follows:

for every $i\in J$. By definition, one has ${\eta }_{{H}_{d,i}}\notin {H}_{d,i}$, as well as

${\eta }_{{H}_{d,i}}{H}_{d,i}{\eta }_{{H}_{d,i}}^{-1}={H}_{d,i}\mathit{ }\text{and}\mathit{ }{\eta }_{{H}_{d,i}}^{2}={\text{id}}_{{ℝ}^{d}}$

for every $i\in J$.

Moreover, for every $i\in J$, let us consider the compact group

${H}_{d,{\eta }_{i}}:=〈{H}_{d,i},{\eta }_{{H}_{d,i}}〉,$

that is, ${H}_{d,{\eta }_{i}}={H}_{d,i}\cup {\eta }_{{H}_{d,i}}{H}_{d,i}$, and the action $⊛:{H}_{d,{\eta }_{i}}×{E}_{V}\to {E}_{V}$ of ${H}_{d,{\eta }_{i}}$ on ${E}_{V}$ is given by

(4.1)

for every $x\in {ℝ}^{d}$. We notice that $⊛$ is defined for every element of ${H}_{d,{\eta }_{i}}$. Indeed, if $h\in {H}_{d,{\eta }_{i}}$, then either $h\in {H}_{d,i}$ or $h=\tau g\in {H}_{d,{\eta }_{i}}\setminus {H}_{d,i}$, with $g\in {H}_{d,i}$. Moreover, set

for every $i\in J$. Following Bartsch and Willem [13], for every $i\in J$, the embedding

${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)↪{L}^{q}\left({ℝ}^{d}\right)$

is compact, for every $q\in \left(2,{2}^{*}\right)$.

Finally, the following facts hold:

• If $d=4$ or $d\ge 6$, then

${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)\cap {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)=\left\{0\right\}$(4.2)

for every $i\in J$.

• If $d=6$ or $d\ge 8$, then

${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)\cap {\mathrm{Fix}}_{{H}_{d,{\eta }_{j}}}\left({E}_{V}\right)=\left\{0\right\}$(4.3)

for every $i,j\in J$ and $i\ne j$.

See [29, Theorem 2.2] for details.

#### Remark 10.

We notice that, if we consider the elliptic problem

requiring that ${V}_{{H}_{d,{\eta }_{i}}}$ and ${W}_{{H}_{d,{\eta }_{i}}}$ be ${H}_{d,{\eta }_{i}}$-invariant (instead of radially symmetric) under the action of the group ${H}_{d,{\eta }_{i}}$ on ${ℝ}^{d}$, for some $i\in J$, then Theorem 6 ensures the existence of at least one non-trivial solution.

Let

${c}_{i,q}:=sup\left\{\frac{{\parallel u\parallel }_{q}}{{\parallel u\parallel }_{{E}_{V}}}:u\in {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)\setminus \left\{0\right\}\right\}$

for every $q\in \left(2,{2}^{*}\right)$, and $i\in J$. Setting

${\lambda }_{i,q}^{\star }:=\frac{q}{{\alpha }_{f}{c}_{i,q}}\underset{\gamma >0}{\mathrm{max}}\left(\frac{\gamma }{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{i,q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\gamma }^{q-1}}\right),$(4.4)

our main result reads as follows.

#### Theorem 11.

Assume $d\mathrm{>}\mathrm{3}$ and let f, V and W as in Theorem 6. In addition, suppose that the nonlinearity f is odd. Then there exists a positive number ${\lambda }_{\mathrm{\star }}$ given by

such that, for every $\lambda \mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}{\lambda }_{\mathrm{\star }}\mathrm{\right)}$, the problem

(Slambda’)

${\zeta }_{S}^{\left(d\right)}:=1+{\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of non-trivial weak solutions ${\mathrm{\left\{}\mathrm{±}{u}_{\lambda \mathrm{,}i}\mathrm{\right\}}}_{i\mathrm{\in }{J}^{\mathrm{\prime }}}\mathrm{\subset }{E}_{V}$, where ${J}^{\mathrm{\prime }}\mathrm{:=}\mathrm{\left\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}{\zeta }_{S}^{\mathrm{\left(}d\mathrm{\right)}}\mathrm{\right\}}$, such that

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda ,i}\parallel }_{{E}_{V}}=0,$

and $\mathrm{|}{u}_{\lambda \mathrm{,}i}\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{|}\mathrm{\to }\mathrm{0}$, as $\mathrm{|}x\mathrm{|}\mathrm{\to }\mathrm{\infty }$, for every $i\mathrm{\in }{J}^{\mathrm{\prime }}$. Moreover, if $d\mathrm{\ne }\mathrm{5}$, problem (Slambda’) admits at least

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of sign-changing weak solutions ${\mathrm{\left\{}\mathrm{±}{z}_{\lambda \mathrm{,}i}\mathrm{\right\}}}_{i\mathrm{\in }J}\mathrm{\subset }{E}_{V}$.

#### Proof.

We divide the proof into two parts.

Part 1: Dimension $d\mathrm{=}\mathrm{5}$. Since f is odd, the energy functional

is even. Owing to Theorem 6, for every $\lambda \in \left(0,{\lambda }^{\star }\right)$, problem (Slambda’) admits at least one (that is, ${\zeta }_{S}^{\left(5\right)}=1$) non-trivial pair of radial weak solutions $\left\{±{u}_{\lambda }\right\}\subset {E}_{V}$. Furthermore, the functions $±{u}_{\lambda }$ are homoclinic and

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=0.$

This concludes the first part of the proof.

Part 2: Dimension $d\mathrm{>}\mathrm{3}$ and $d\mathrm{\ne }\mathrm{5}$. For every $\lambda >0$ and $i=1,2,\mathrm{\dots },{\tau }_{d}$, consider the restrictions

${\mathcal{ℋ}}_{\lambda ,i}:={{J}_{\lambda }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}:{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)\to ℝ$

defined by

${\mathcal{ℋ}}_{\lambda ,i}:={\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}\left(u\right)-{\lambda \mathrm{\Psi }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}\left(u\right),$

where

${\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}\left(u\right):=\frac{1}{2}{\parallel u\parallel }_{{E}_{V}}^{2}\mathit{ }\text{and}\mathit{ }{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}\left(u\right):={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left(x\right)\right)𝑑x$

for every $u\in {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$. In order to obtain the existence of

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of sign-changing weak solutions ${\left\{±{z}_{\lambda ,i}\right\}}_{i\in J}\subset {E}_{V}$, where $J:=\left\{1,\mathrm{\dots },{\tau }_{d}\right\}$, the main idea of the proof consists in applying Theorem 3 to the functionals ${\mathcal{ℋ}}_{\lambda ,i}$ for every $i\in J$. We notice that, since $d>3$ and $d\ne 5$, ${\tau }_{d}\ge 1$. Consequently, the cardinality $|J|\ge 1$.

Since $0<\lambda <{\lambda }_{i,q}^{\star }$, with $i\in J$, there exists ${\overline{\gamma }}_{i}>0$ such that

$\lambda <{\lambda }_{\star }^{\left(i\right)}\left({\overline{\gamma }}_{i}\right):=\frac{q}{{\alpha }_{f}{c}_{i,q}}\left(\frac{{\overline{\gamma }}_{i}}{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{i,q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}_{i}^{q-1}}\right).$

Similar arguments used proving (3.7) yield

$\phi \left({\overline{\gamma }}_{i}^{2}\right)\le \chi \left({\overline{\gamma }}_{i}^{2}\right)\le {\alpha }_{f}{c}_{q}\left(\sqrt{2}\frac{{\parallel W\parallel }_{p}}{{\overline{\gamma }}_{i}}+\frac{{2}^{\frac{q}{2}}{c}_{q}^{q-1}}{q}{\parallel W\parallel }_{\mathrm{\infty }}\overline{\gamma }_{i}{}^{q-2}\right)<\frac{1}{\lambda }.$

Thus,

$\lambda \in \left(0,\frac{q}{{\alpha }_{f}{c}_{q}}\left(\frac{\overline{\gamma }}{q\sqrt{2}{\parallel W\parallel }_{p}+{2}^{\frac{q}{2}}{c}_{q}^{q-1}{\parallel W\parallel }_{\mathrm{\infty }}{\overline{\gamma }}^{q-1}}\right)\right)\subseteq \left(0,\frac{1}{\phi \left({\overline{\gamma }}_{i}^{2}\right)}\right).$

Thanks to Theorem 3, there exists a function ${z}_{\lambda ,i}\in {\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{i}^{2}\right)\right)$ such that

${\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{\prime }\left({z}_{\lambda ,i}\right)-\lambda {\left({\mathrm{\Psi }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}\right)}^{\prime }\left({z}_{\lambda ,i}\right)=0,$

and, in particular, ${z}_{\lambda ,i}$ is a global minimum of the restriction of ${\mathcal{ℋ}}_{\lambda ,i}$ to ${\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{i}^{2}\right)\right)$.

Due to the evenness of ${J}_{\lambda }$, bearing in mind (4.1), and thanks to the symmetry assumptions on the potentials V and W, we have that ${J}_{\lambda }\left(h⊛u\right)={J}_{\lambda }\left(u\right)$ for every $h\in {H}_{d,{\eta }_{i}}$ and $u\in {E}_{V}$, i.e., the functional ${J}_{\lambda }$ is ${H}_{d,{\eta }_{i}}$-invariant on ${E}_{V}$. Indeed, ${H}_{d,{\eta }_{i}}$ acts isometrically on ${E}_{V}$ (note that V is radial) and, thanks to the symmetry assumption on W, one has

${\int }_{{ℝ}^{d}}W\left(x\right)F\left(h⊛u\left(x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left({h}^{-1}x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(y\right)F\left(u\left(y\right)\right)𝑑y$

if $h\in {H}_{d,i}$, and

${\int }_{{ℝ}^{d}}W\left(x\right)F\left(h⊛u\left(x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(x\right)F\left(u\left({g}^{-1}{\eta }_{{H}_{d,i}}^{-1}x\right)\right)𝑑x={\int }_{{ℝ}^{d}}W\left(y\right)F\left(u\left(y\right)\right)𝑑y$

if $h={\eta }_{{H}_{d,i}}g\in {H}_{d,{\eta }_{i}}\setminus {H}_{d,i}$.

On account of the principle of symmetric criticality (recalled in Theorem 4), the critical point pairs $\left\{±{z}_{\lambda ,i}\right\}$ of ${\mathcal{ℋ}}_{\lambda ,i}$ are also critical points of ${J}_{\lambda }$. Now, we have to show that the solution ${z}_{\lambda ,i}$ found here above is not the trivial function. If $f\left(0\right)\ne 0$, then it easily follows that ${z}_{\lambda ,i}\not\equiv 0$ in ${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$, since the trivial function does not solve problem (Slambda’). Let us consider the case when $f\left(0\right)=0$ and let ${z}_{\lambda ,i}$ be such that

and

${\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}\left({z}_{\lambda ,i}\right)<{\overline{\gamma }}_{i}^{2},$

and also ${z}_{\lambda ,i}$ is a critical point of ${\mathcal{ℋ}}_{\lambda ,i}$ in ${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$. In this setting, in order to prove that we have ${z}_{\lambda ,i}\not\equiv 0$ in ${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$, first we claim that there exists a sequence ${\left\{{w}_{j}^{i}\right\}}_{j\in ℕ}$ in ${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$ such that

$\underset{j\to +\mathrm{\infty }}{lim sup}\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}\left({w}_{j}^{i}\right)}{\mathrm{\Phi }\left({w}_{j}^{i}\right)}=+\mathrm{\infty }.$(4.5)

In order to construct the sequence ${\left\{{w}_{j}^{i}\right\}}_{j\in ℕ}\subset {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$ for which (4.5) holds, we use, in a suitable way, the test functions introduced by Kristály, Moroşanu and O’Regan in [29]. Let $0 be such that condition (3.11) holds and $r\ge \frac{R}{5+4\sqrt{2}}$. Set $\sigma \in \left(0,1\right)$ and define ${v}_{\sigma }^{i}\in {E}_{V}$ as follows:

for every $x\in {ℝ}^{d}$, where

${v}_{\sigma }^{\frac{d-2}{2}}\left({x}_{1},{x}_{3}\right):=\left[{\left(\frac{R-r}{4}-\mathrm{max}\left\{\sqrt{{\left({|{x}_{1}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{3}|}^{2}},\sigma \frac{R-r}{4}\right\}\right)}_{+}$$-{\left(\frac{R-r}{4}-\mathrm{max}\left\{\sqrt{{\left({|{x}_{1}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{3}|}^{2}},\sigma \frac{R-r}{4}\right\}\right)}_{+}\right]×\frac{4}{\left(R-r\right)\left(1-\sigma \right)}$

for every $\left({x}_{1},{x}_{3}\right)\in {ℝ}^{\frac{d}{2}}×{ℝ}^{\frac{d}{2}}$ and

${v}_{i}^{\sigma }\left({x}_{1},{x}_{2},{x}_{3}\right):=\left[{\left(\frac{R-r}{4}-\mathrm{max}\left\{\sqrt{{\left({|{x}_{1}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{3}|}^{2}},\sigma \frac{R-r}{4}\right\}\right)}_{+}$$-{\left(\frac{R-r}{4}-\mathrm{max}\left\{\sqrt{{\left({|{x}_{3}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{1}|}^{2}},\sigma \frac{R-r}{4}\right\}\right)}_{+}\right]$$×{\left(\frac{R-r}{4}-\mathrm{max}\left\{|{x}_{2}|,\sigma \frac{R-r}{4}\right\}\right)}_{+}\frac{4}{{\left(R-r\right)}^{2}{\left(1-\sigma \right)}^{2}}$

for every $\left({x}_{1},{x}_{2},{x}_{3}\right)\in {ℝ}^{\frac{d}{2}}×{ℝ}^{d-2i-2}×{ℝ}^{\frac{d}{2}}$, and $i\ne \frac{d-2}{2}$. Now, it is possible to prove that ${v}_{\sigma }^{i}\in {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$. Moreover, for every $\varrho \in \left(0,1\right]$, let

${Q}_{\varrho }^{\left(1\right)}:=\left\{\left({x}_{1},{x}_{3}\right)\in {ℝ}^{i+1}×{ℝ}^{i+1}:\sqrt{{\left({|{x}_{1}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{3}|}^{2}}\le \varrho \frac{R-r}{4}\right\},$${Q}_{\varrho }^{\left(2\right)}:=\left\{\left({x}_{1},{x}_{3}\right)\in {ℝ}^{i+1}×{ℝ}^{i+1}:\sqrt{{\left({|{x}_{3}|}^{2}-\frac{R+3r}{4}\right)}^{2}+{|{x}_{1}|}^{2}}\le \varrho \frac{R-r}{4}\right\}.$

Define

where

${D}_{\varrho }^{\frac{d-2}{2}}:=\left\{\left({x}_{1},{x}_{3}\right)\in {ℝ}^{\frac{d}{2}}×{ℝ}^{\frac{d}{2}}:\left({x}_{1},{x}_{3}\right)\in {Q}_{\varrho }^{\left(1\right)}\cap {Q}_{\varrho }^{\left(2\right)}\right\},$

for every $i\ne \frac{d-2}{2}$. The sets ${D}_{\varrho }^{i}$ have positive Lebesgue measure and they are ${H}_{d,{\eta }_{i}}$-invariant. Moreover, for every $\sigma \in \left(0,1\right)$, one has ${v}_{\sigma }^{i}\in {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$ and the following facts hold:

• (1)

$\mathrm{supp}\left({v}_{\sigma }^{i}\right)={D}_{1}^{i}\subseteq A\left[r,R\right]$,

• (2)

${\parallel {v}_{\sigma }^{i}\parallel }_{\mathrm{\infty }}\le 1$,

• (3)

$|{v}_{\sigma }^{i}\left(x\right)|=1$ for every $x\in {D}_{\sigma }^{i}$.

Thus, let ${w}_{j}^{i}:={t}_{j}{v}_{\sigma }^{i}$ for any $j\in ℕ$. Of course, ${w}_{j}^{i}\in {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$ for any $j\in ℕ$. Furthermore, taking into account the properties of ${v}_{\sigma }^{i}$ stated in 13, since F is even (this implies that $F\left({w}_{j}^{i}\left(x\right)\right)=F\left({t}_{j}\right)$ for every $x\in {D}_{\sigma }^{i}$) with $F\left(0\right)=0$, and by using (3.10) one has for j sufficiently large,

$\frac{{\mathrm{\Psi }|}_{{\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)}\left({w}_{j}^{i}\right)}{\mathrm{\Phi }\left({w}_{j}^{i}\right)}=\frac{{\int }_{{D}_{\sigma }^{i}}W\left(x\right)F\left({w}_{j}^{i}\left(x\right)\right)𝑑x+{\int }_{{A}_{r}^{R}\setminus {D}_{\sigma }^{i}}W\left(x\right)F\left({w}_{j}^{i}\left(x\right)\right)𝑑x}{\mathrm{\Phi }\left({w}_{j}^{i}\right)}$$=\frac{{\int }_{{D}_{\sigma }^{i}}W\left(x\right)F\left({t}_{j}\right)𝑑x+{\int }_{{A}_{r}^{R}\setminus {D}_{\sigma }^{i}}W\left(x\right)F\left({t}_{j}{v}_{\sigma }^{i}\left(x\right)\right)𝑑x}{\mathrm{\Phi }\left({w}_{j}^{i}\right)}$$\ge 2\frac{M|{D}_{\sigma }^{i}|\alpha {t}_{j}^{2}+{\int }_{{A}_{r}^{R}\setminus {D}_{\sigma }^{i}}W\left(x\right)F\left({t}_{j}{v}_{\sigma }^{i}\left(x\right)\right)𝑑x}{{t}_{j}^{2}{\parallel {v}_{\sigma }^{i}\parallel }_{{E}_{V}}^{2}}.$(4.6)

Arguing as in the proof of Theorem 6, inequality (4.6) yields (4.5) and consequently we conclude that

${\mathcal{ℋ}}_{\lambda ,i}\left({z}_{\lambda ,i}\right)\le {\mathcal{ℋ}}_{\lambda ,i}\left({w}_{j}^{i}\right)<0={\mathcal{ℋ}}_{\lambda ,i}\left(0\right),$

so that ${z}_{\lambda ,i}\not\equiv 0$ in ${\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right)$. In addition, by adapting again the arguments used along the proof of Theorem 6 it follows that

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {z}_{\lambda ,i}\parallel }_{{E}_{V}}=0,$

and $|{z}_{\lambda ,i}\left(x\right)|\to 0$ as $|x|\to \mathrm{\infty }$.

On the other hand, since $\lambda <{\lambda }^{\star }$ and f is odd, Theorem 6 and the principle of symmetric criticality (recalled in Theorem 4) ensure that problem (Slambda’) admits at least one non-trivial pair of radial weak solutions $\left\{±{u}_{\lambda }\right\}\subset {E}_{V}$. Moreover,

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=0,$

and $|{u}_{\lambda }\left(x\right)|\to 0$ as $|x|\to \mathrm{\infty }$.

In conclusion, since $\lambda <{\lambda }_{\star }$, there exist ${\tau }_{d}+1$ positive numbers $\overline{\gamma }$, ${\overline{\gamma }}_{1},\mathrm{\dots },{\overline{\gamma }}_{{\tau }_{d}}$ such that

$±{u}_{\lambda }\in {\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)\setminus \left\{0\right\}\subset {\mathrm{Fix}}_{O\left(d\right)}\left({E}_{V}\right)\mathit{ }\text{and}\mathit{ }±{z}_{\lambda ,i}\in {\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{2}^{2}\right)\right)\setminus \left\{0\right\}\subset {\mathrm{Fix}}_{{H}_{d,{\eta }_{i}}}\left({E}_{V}\right).$

Bearing in mind relations (4.2) and (4.3) (see also [29, Theorem 2.2] for details) we have that

${\mathrm{\Phi }}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}^{2}\right)\right)\cap {\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{i}^{2}\right)\right)\setminus \left\{0\right\}=\mathrm{\varnothing }$

for every $i\in J$ and

${\mathrm{\Phi }}_{{H}_{d,{\eta }_{i}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{i}^{2}\right)\right)\cap {\mathrm{\Phi }}_{{H}_{d,{\eta }_{j}}}^{-1}\left(\left(-\mathrm{\infty },{\overline{\gamma }}_{j}^{2}\right)\right)\setminus \left\{0\right\}=\mathrm{\varnothing }$

for every $i,j\in J$ and $i\ne j$. Consequently, problem (Slambda’) admits at least

${\zeta }_{S}^{\left(d\right)}:={\tau }_{d}+1$

pairs of non-trivial weak solutions ${\left\{±{u}_{\lambda ,i}\right\}}_{i\in {J}^{\prime }}\subset {E}_{V}$, where ${J}^{\prime }:=\left\{1,\mathrm{\dots },{\zeta }_{S}^{\left(d\right)}\right\}$, such that

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda ,i}\parallel }_{{E}_{V}}=0,$

and $|{u}_{\lambda ,i}\left(x\right)|\to 0$, as $|x|\to \mathrm{\infty }$, for every $i\in {J}^{\prime }$. Moreover, by construction, it follows that

${\tau }_{d}:={\left(-1\right)}^{d}+\left[\frac{d-3}{2}\right]$

pairs of the attained solutions are sign-changing.

The proof is now complete. ∎

#### Remark 12.

In order to obtain a concrete form of the interval of parameters for which our results hold, it is necessary an explicit computation of the Sobolev embedding constants that naturally appear in Theorem 11 as well as in Theorem 6.

#### Remark 13.

We notice that the statement of Theorem 11 is not relevant in dimension three. However, also in this case, Theorem 11 gives one distinct (pair of) non-trivial and radially symmetric solution for (Slambda) whenever λ is sufficiently small.

#### Remark 14.

The conclusions of Theorem 1 in the Introduction immediately follow by Theorem 11 provided $\lambda \in \left(0,{\lambda }_{\star }\right)$.

We end this paper by exhibiting the example of a nonlinearity satisfying Theorem 1 together with the related estimate of the parameter λ.

#### Example 15.

Let $1 and $2 and let V be a potential for which $\left({h}_{V}^{4}\right)$ holds. Then, owing to Theorem 11, for λ sufficiently small, the problem

(Plambda)

admits at least one non-trivial pair of radially symmetric weak solutions $\left\{±{u}_{\lambda }\right\}$ and one pair of sign-changing weak solutions $\left\{±{z}_{\lambda }\right\}$. Moreover,

$\underset{\lambda \to {0}^{+}}{lim}{\parallel {u}_{\lambda }\parallel }_{{E}_{V}}=\underset{\lambda \to {0}^{+}}{lim}{\parallel {z}_{\lambda }\parallel }_{{E}_{V}}=0,$

and $|{u}_{\lambda }\left(x\right)|\to 0$, as well as $|{z}_{\lambda }\left(x\right)|\to 0$, as $|x|\to \mathrm{\infty }$. More precisely, set

${\parallel W\parallel }_{{s}^{\prime }}:={\left({\int }_{{ℝ}^{4}}\frac{1}{{\left(1+{|x|}^{4}\right)}^{2}}𝑑x\right)}^{\frac{s-1}{s}}={\left(\frac{\mathrm{area}\left({𝕊}^{3}\right)}{4}\right)}^{\frac{s-1}{s}},$

where ${𝕊}^{3}$ denotes the 4-dimensional Euclidean unit sphere and

${\kappa }_{s}:=sup\left\{\frac{{\parallel u\parallel }_{q}}{{\parallel u\parallel }_{{E}_{V}}}:u\in {\mathrm{Fix}}_{{H}_{4,\tau }}\left({E}_{V}\right)\setminus \left\{0\right\}\right\},$

where $\tau :{ℝ}^{4}\to {ℝ}^{4}$ is the involution given by $\tau \left({x}_{1},{x}_{3}\right):=\left({x}_{3},{x}_{1}\right)$ for every $\left({x}_{1},{x}_{3}\right)\in {ℝ}^{2}×{ℝ}^{2}$ and associated to the subgroup $O\left(2\right)×O\left(2\right)\subset O\left(4\right)$. If

${c}_{s}:=sup\left\{\frac{{\parallel u\parallel }_{s}}{{\parallel u\parallel }_{{E}_{V}}}:u\in {\mathrm{Fix}}_{O\left(4\right)}\left({E}_{V}\right)\setminus \left\{0\right\}\right\},$

then the existence result claimed for problem (Plambda) is valid for every

$0<\lambda <\frac{s}{2}\mathrm{min}\left\{\frac{1}{{\kappa }_{s}}\underset{\gamma >0}{\mathrm{max}}\left(\frac{\gamma }{s\sqrt{2}{\parallel W\parallel }_{{s}^{\prime }}+{2}^{\frac{s}{2}}{\kappa }_{s}^{s-1}{\gamma }^{s-1}}\right),\frac{1}{{c}_{s}}\underset{\gamma >0}{\mathrm{max}}\left(\frac{\gamma }{s\sqrt{2}{\parallel W\parallel }_{{s}^{\prime }}+{2}^{\frac{s}{2}}{c}_{s}^{s-1}{\gamma }^{s-1}}\right)\right\}.$

#### Remark 16.

In [29] Kristály, Moroşanu and O’Regan studied the (non-)existence of (non-)radial solutions for perturbed Schrödinger equations by using a similar variational approach and under suitable assumptions on the nonlinear term f. More precisely, they assume that the datum f satisfies the following conditions:

• (h1)

$f\left(t\right)=o\left(|t|\right)$ as $|t|\to \mathrm{\infty }$,

• (h2)

$f\left(t\right)=o\left(|t|\right)$ as $|t|\to 0$,

• (h3)

there exists ${t}_{0}\in ℝ$ such that $F\left({t}_{0}\right)>0$.

We emphasize that our results are mutually independent. For instance, [29, Theorem 1.1] cannot be applied to problem (Plambda) studied in Example 15. Indeed, in this case, since $r<2$, one has

$\underset{t\to {0}^{+}}{lim}\frac{f\left(t\right)}{t}=\underset{t\to {0}^{+}}{lim}\frac{{t}^{r-1}+{t}^{s-1}}{t}=+\mathrm{\infty },$

and consequently $\left({h}_{2}\right)$ is not verified.

#### Remark 17.

Theorems 6 and 11 are a paradigmatic application of Theorem 3 for elliptic partial differential equations on unbounded domains. This fact is due to the peculiar nature of the conclusion of the abstract Ricceri’s result. Indeed, the interplay between Theorem 3 with the Palais Principle (see Theorem 4) turns out to be successful in our setting thanks to the preliminary results presented in Sections 2 and 4.

#### Remark 18.

For completeness we mention here some recent contributions on elliptic problems defined on unbounded domains and related to the results contained in this paper [8, 22, 32, 33]. Finally, we point out that some of the theorems presented in this paper could be also achieved for a larger class of elliptic equations where the leading term is governed by some differential operators considered in [6, 7, 5]. However, in this cases, some different technical approaches need to be adopted in order to arrive to the analogous desired existence results for this wider class of energies. We will consider this interesting case in some further investigations.

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Accepted: 2018-06-05

Published Online: 2018-07-04

The paper is realized with the auspices of the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009) and the INdAM-GNAMPA Project 2017 titled Teoria e modelli non-locali.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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