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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 d (d3). 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(d). In addition, if the nonlinear term is odd, and d>3, the existence of (-1)d+[d-32] pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the L-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.

Keywords: Schrödinger equation; variational methods; principle of symmetric criticality; radial and non-radial solutions

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

1 Introduction

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

{-Δu+V(x)u=λW(x)f(u)in d,uH1(d),(Slambda)

where (d,||) is the Euclidean space (with d3), Δu:=div(u) stands for the classical Laplacian operator, f: is a subcritical continuous function, WL(N)L2(d){0} 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:d satisfies the following condition:

  • ${(h_{V}^{d})}$

    VC0(d) is radially symmetric and infxdV(x)>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

ut=Δu-V(x)u+λW(x)f(u),

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:

{-Δu+Vu=Wf(u)in d,uH1(d).(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 d2+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 d6 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

{-Δu+V(x)u=λW(x)(|u|r-2u+|u|s-2u)in d,uH1(d),

where 1<r<2<s<2dd-2, 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>3 and let 1<r<2<s<2*, where 2*:=2dd-2. Furthermore, let t be the conjugate exponent of s and let WL(Rd)Lt(Rd){0} be a non-negative radially symmetric map. Finally, let V be a potential for which (hVd) holds. Then, for λ sufficiently small, the problem

{-Δu+V(x)u=λW(x)(|u|r-2u+|u|s-2u)in d,uH1(d),(Clambda)

admits at least

ζS(d):=1+(-1)d+[d-32]

pairs of non-trivial weak solutions {±uλ,i}iJEV, where J:={1,,ζS(d)}, such that

limλ0+(d|uλ,i(x)|2𝑑x+dV(x)|uλ,i(x)|2𝑑x)=0,

and |uλ,i(x)|0, as |x|, for every iJ. Moreover, if d5, problem (Clambda) admits at least

τd:=(-1)d+[d-32]

pairs of sign-changing weak solutions {±zλ,i}iJEV, where J:={1,,τd}.

Hereafter, [x] denotes the integer part of a real number x0.

More precisely, in Theorem 6 we prove that there exists a well-localized interval of positive real parameters (0,λ) such that, for every λ(0,λ), problem (Slambda) admits at least one non-trivial radial weak solution in a suitable Sobolev space EV. By a Strauss-type estimate (see Lions [40]) every weak solution uλEV is also homoclinic, that is,

|uλ(x)|0,as |x|.

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

Jλ(u):=12(d|u(x)|2𝑑x+dV(x)|u(x)|2𝑑x)-λdW(x)(0u(x)f(z)𝑑z)𝑑x

for every uEV, 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

lim supt0+0tf(z)𝑑zt2=+,andlim inft0+0tf(z)𝑑zt2>-,

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 Jλ is invariant under certain subgroups actions of the orthogonal group O(d). Due to Theorem 3, every restriction of Jλ 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 Jλ, and they are non-radial solutions of problem (Slambda), depending on the choice of the subgroup of O(d); 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

ζS(d):=1+(-1)d+[d-32]

subspaces of EV. Moreover, when d5, there are

τd:=(-1)d+[d-32]

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,

λ<λ:={λif d=5,min{λ,λi,q:i=1,,τd}if d5,

where λ* and λi,q are suitable geometrical constants whose expression is given respectively in formulas (3.2) and (4.4). The structure of the parameters λ and λ 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:

limz0f(z)z=0.

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

f(t):=|t|r-2t+|t|s-2tfor all t,

where 1<r<2<s<2*, 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 EV; 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 d3 and let

EV:={uH1(d):d(|u(x)|2+V(x)|u(x)|2)𝑑x<+}

be the Hilbert space endowed by the inner product

u,v:=du(x)v(x)𝑑x+dV(x)u(x)v(x)𝑑xfor all u,vEV

and induced norm

uEV:=(d|u(x)|2𝑑x+dV(x)|u(x)|2𝑑x)12

for every uEV.

Thanks to assumption (hV) the space EV is continuously embedded in Lq(d) for every q[2,2*], where 2*:=2dd-2 is the classical critical Sobolev exponent. As pointed out in Introduction, one the main difficulties in working in the space EV is the lack of compactness of these embeddings.

Now, we show that an Lp-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:RR is a continuous function such that

αf:=supz|f(z)|1+|z|q-1<+,(2.1)

where q[2,2*] and let Ψ:EVR be defined by

Ψ(u):=dW(x)F(u(x))𝑑xfor all uEV,

where WL(Rd)+Lp(Rd), where p:=qq-1 is the conjugate Sobolev exponent of q, and F(t):=0tf(z)𝑑z for every tR. Then Ψ is continuously Gâteaux derivable on EV.

Proof.

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

dW(x)F(u(x))𝑑xαfdW(x)|u(x)|𝑑x+αfqdW(x)|u(x)|q𝑑xαf(d|W(x)|p𝑑x)1p(d|u(x)|q𝑑x)1q+αfqWd|u(x)|q𝑑x(2.2)

for every uEV. Since EV is continuously embedded in Lq(d), inequality (2.2) yields

Ψ(u)αf(Wpuq+1qWuqq)<+

for every uEV.

Now, let us compute the Gâteaux derivative Ψ:EVEV* such that uΨ(u),v, where

Ψ(u),v:=limh0Ψ(u+hv)-Ψ(u)h=limh0dW(x)(F(u(x)+hv(x))-F(u(x)))𝑑xh.

We claim that

limh0+dW(x)(F(u(x)+hv(x))-F(u(x)))𝑑xh=dW(x)f(u(x))v(x)𝑑x.(2.3)

Indeed, for a.e. xd, and |h|(0,1), by the Mean Value Theorem, there exists θ(0,1) (depending of x) such that (up to the constant αf)

W(x)|f(u(x)+θhv(x))||v(x)|W(x)(1+|u(x)+θhv(x)|q-1)|v(x)|W(x)|v(x)|+W||u(x)|+|v(x)||q-1|v(x)|W(x)|v(x)|+2q-2W(|u(x)|q-1+|v(x)|q-1)|v(x)|.

Setting

g(x):=W(x)|v(x)|+2q-2W(|u(x)|q-1+|v(x)|q-1)|v(x)|

for a.e. xd, owing to q[2,2*], the Hölder inequality ensures that

dg(x)𝑑xWpvq+2q-2Wuqq-1vq+2q-2Wvqq<+.

Thus gL1(d) and Lebesgue’s Dominated Convergence Theorem ensures that (2.3) holds true.

We prove now the continuity of the Gâteaux derivative Ψ. Assume that uju in EV as j+, and let us prove that

limj+Ψ(uj)-Ψ(u)EV*=0,

where

Ψ(uj)-Ψ(u)EV*:=supvEV=1|Ψ(uj)-Ψ(u),v|.

Observe that

Ψ(uj)-Ψ(u)EV*:=supvEV=1|Ψ(uj)-Ψ(u),v|supvEV=1|dW(x)(f(uj(x))-f(u(x)))v(x)𝑑x|supvEV=1dW(x)|(f(uj(x))-f(u(x)))||v(x)|𝑑x.(2.4)

The Sobolev imbedding result ensures that uju in Lq(n) for every q[2,2*], and uj(x)u(x) a.e. in d as j+. Then, by [42, Lemma A.1], up to a subsequence, uj(x)u(x) a.e. in d as j+, and there exists τLq(d) such that |uj(x)|τ(x) a.e. in d and |u(x)|τ(x) a.e. in d. In particular,

f(uj(x))f(u(x))a.e. in d,|f(uj(x))|αf(1+τ(x)q-1)a.e. in d.

Note that the function dx1+τ(x)q-1 is p:=qq-1-summable on every bounded measurable subset of d. Therefore, by the Lebesgue Dominated Convergence Theorem, we infer

f(uj)f(u)in Lp(BR(0)),

for every R>0, where BR(0) denotes the open ball of radius R centered at zero. At this point, set

q:=supvEV1vq

and let Rε>0 be such that

dBRε(0)|W(x)|p𝑑x<(ε8αfq)p,(2.5)

as well as

dBRε(0)|τ(x)|p𝑑x<(ε8αfqW)p.(2.6)

Moreover, let jε be such that

f(uj)-f(u)Lp(BRε(0))<ε4qW(2.7)

for each j, with jjε. Owing to (2.5), (2.6) and (2.7), for jjε, inequality (2.4) yields

Ψ(uj)-Ψ(u)EV*supvEV1dBRε(0)W(x)|f(uj(x))-f(u(x))||v(x)|𝑑x2αfsupvEV1dBRε(0)W(x)(1+|τ(x)|q-1)|v(x)|𝑑x+WsupvEV1BRε(0)|f(uj(x))-f(u(x))||v(x)|𝑑x2αfqWLp(dBRε(0))+2αfqWτLp(dBRε(0))+qWf(uj)-f(u)Lp(BRε(0))<ε,

that concludes the proof. ∎

Of course, by elementary standard arguments, Proposition 2 ensures that ΨC1(EV,). Then the functional defined by

Jλ:=12uEV2-λdW(x)F(u(x))𝑑xfor all uEV

is of class C1(EV,). Furthermore, the critical points of Jλ are exactly the weak solutions of problem (Slambda). More precisely, we say that uEV is a weak solution of problem (Slambda) if and only if

du(x)φ(x)𝑑x+dV(x)u(x)φ(x)𝑑x=λdW(x)f(u(x))φ(x)𝑑xfor all φEV.

In order to find critical points for Jλ, 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 Φ,Ψ:XR 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>infXΦ, put

φ(r):=infuΦ-1((-,r))(supvΦ-1((-,r))Ψ(v))-Ψ(u)r-Φ(u).

Then, for each r>infXΦ and each λ(0,1φ(r)), the restriction of Jλ:=Φ-λΨ to Φ-1((-,r)) admits a global minimum, which is a critical point (local minimum) of Jλ 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 (X,X) is a continuous map

*:G×XX,(g,x)g*u

such that

1*u=u,(g*h)u=g*(h*u),ug*u is linear.

The action * is said to be isometric if g*uX=uX for every gG. Moreover, the space of G-invariant points is defined by

FixG(X):={uX:g*u=u for all gG},

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

h(g*u)=h(u)

for every gG and uX.

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

Assume that the action of the topological group G on the Banach space X is isometric. If IC1(X,R) is G-invariant on X and if u is a critical point of I restricted to FixG(X), then u is a critical point of I.

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

Let O(d) be the orthogonal group and let GO(d) be a subgroup. Assume that G acts on the space EV. Hence, the set of fixed points of EV, respect to O(d), is clearly given by

FixO(d)(X):={uEV:gu=u for all gO(d)}.

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

gu(x):=u(g-1x)for all xd and gO(d),

then FixO(d)(EV) is exactly the subspace of radially symmetric functions of EV. Moreover, the embedding

FixO(d)(EV)Lq(d)(2.8)

is continuous (respectively, compact) for every q[2,2*] (respectively, q(2,2*)). See, for instance, the celebrated paper [31].

Now, for every λ>0, let 𝒥λ:=Jλ|FixO(d)(EV):FixO(d)(EV) be the functional defined by

𝒥λ(u):=Φ(u)-λΨ|FixO(d)(EV)(u),

where

Φ(u):=12uEV2andΨ(u):=dW(x)F(u(x))𝑑xfor all uFixO(d)(EV).

The above remarks yield the next semicontinuity property.

Lemma 5.

Assume that f:RR is a continuous function such that condition (2.1) holds for every q(2,2*). Then, for every λ>0, the functional Jλ is sequentially weakly lower semicontinuous on FixO(d)(EV).

Proof.

First, on account of Brézis [16, Corollaire III.8], the functional Φ is sequentially weakly lower semicontinuous on FixO(d)(EV). In order to prove that Ψ|FixO(d)(EV) is sequentially weakly continuous, we assume that there exists a sequence {uj}jFixO(d)(EV) which weakly converges to an element u0FixO(d)(EV). Since {uj}j is bounded in FixO(d)(EV) and, taking into account that, thanks to (2.8), uju0 in Lq(d), the Mean Value Theorem, the growth condition (2.1) and the Hölder inequality yield

|Ψ(uj)-Ψ(u0)|dW(x)|F(uj(x))-F(u0(x))|𝑑xαfdW(x)(2+|uj(x)|q-1+|u0(x)|q-1)|uj(x)-u0(x)|𝑑xαf(2Wpuj-u0q+W(ujqq-1+u0qq-1)uj-u0q)αf(2W2+W(M+u0qq-1))uj-u0q(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 d3 and set

c:=sup{uuEV:uFixO(d)(EV){0}}

for every (2,2*).

With the above notation the main result reads as follows.

Theorem 6.

Let f:RR be a continuous function satisfying the growth condition (2.1) for some q(2,2*) in addition to

lim supt0+F(t)t2=+𝑎𝑛𝑑lim inft0+F(t)t2>-,(3.1)

where F(t):=0tf(z)𝑑z. Furthermore, let WL(Rd)Lp(Rd){0}, be a radially symmetric map with

W(x)0in d,

and V be a potential for which (hVd) holds. Then there exists a positive number λ given by

λ:=qαfcqmaxγ>0(γq2Wp+2q2cqq-1Wγq-1)(3.2)

such that, for every λ(0,λ), the problem

{-Δu+V(x)u=λW(x)f(u)in d,uH1(d),(Slambda)

admits at least one non-trivial radial weak solution uλEV. Moreover,

limλ0+uλEV=0,

and |uλ(x)|0 as |x|.

Proof.

The main idea of the proof consists in applying Theorem 3 to the functional 𝒥λ. 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 𝒥λ as follows:

𝒥λ(u)=Φ(u)-λΨ|FixO(d)(EV)(u)for all uFixO(d)(EV)

with

Φ(u):=12(d|u(x)|2𝑑x+dV(x)|u(x)|2𝑑x)

as well as

Ψ(u):=dW(x)F(u(x))𝑑x.

First of all, note that FixO(d)(EV) is a Hilbert space and the functionals Φ and Ψ|FixO(d)(EV) have the regularity required by Theorem 3 (see Lemma 5). Moreover, it is clear that the functional Φ is strongly continuous, coercive in FixO(d)(EV) and

infuFixO(d)(EV)Φ(u)=0.

Now, since 0<λ<λ, bearing in mind (3.2), there exists γ¯>0 such that

λ<λ(γ¯):=qαfcq(γ¯q2Wp+2q2cqq-1Wγ¯q-1).(3.3)

Set r(0,+) and consider the function χ:(0,+)[0,+) given by

χ(r):=supuΦ-1((-,r))Ψ|FixO(d)(EV)(u)r.

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

Ψ|FixO(d)(EV)(u)=dW(x)F(u(x))𝑑xαfdW(x)|u(x)|𝑑x+αfqdW(x)|u(x)|q𝑑x.

Moreover, one has

uEV<2r(3.4)

for every uFixO(d)(EV) and Φ(u)<r.

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

Ψ|FixO(d)(EV)(u)<αfcq(Wp2r+cqq-1qW(2r)q2),

for every uFixO(d)(EV) such that Φ(u)<r. Hence

supuΦ-1((-,r))Ψ|FixO(d)(EV)(u)αfcq(Wp2r+cqq-1qW(2r)q2).

Then the above inequality immediately gives

χ(r)αfcq(Wp2r+2q2cqq-1qWrq2-1)(3.5)

for every r>0.

Evaluating inequality (3.5) in r=γ¯2, we have

χ(γ¯2)αfcq(2Wpγ¯+2q2cqq-1qWγ¯q-2).(3.6)

Now, it is easy to note that

φ(γ¯2):=infuΦ-1((-,γ¯2))(supvΦ-1((-,γ¯2))Ψ|FixO(d)(EV)(v))-Ψ|FixO(d)(EV)(u)r-Φ(u)χ(γ¯2),

owing to z0Φ-1((-,γ¯2)) and Φ(z0)=Ψ|FixO(d)(EV)(z0)=0, where z0FixO(d)(EV) is the zero function.

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

φ(γ¯2)χ(γ¯2)αfcq(2Wpγ¯+2q2cqq-1qWγ¯q-2)<1λ.(3.7)

Hence, we have that

λ(0,qαfcq(γ¯q2Wp+2q2cqq-1Wγ¯q-1))(0,1φ(γ¯2)).

The (critical point) Theorem 3 ensures that there exists a function uλΦ-1((-,γ¯2)) such that

Φ(uλ)-λ(Ψ|FixO(d)(EV))(uλ)=0

and, in particular, uλ is a global minimum of the restriction of the functional 𝒥λ to the sublevel Φ-1((-,γ¯2)).

Now, we have to show that the solution uλ found here above is not the trivial (identically zero) function. If f(0)0, then it easily follows that uλ0 in FixO(d)(EV), since the trivial function does not solve problem (Slambda). Let us consider the case when f(0)=0 and let us fix λ(0,λ(γ¯)) for some γ¯>0. Finally, let uλ be such that

𝒥λ(uλ)𝒥λ(u)for any uFixO(d)(EV) such that Φ(u)<γ¯2(3.8)

and

Φ(uλ)<γ¯2,

and also uλ is a critical point of 𝒥λ in FixO(d)(EV). Since O(d) acts isometrically on EV (note that V is radial) and taking into account that, thanks to the symmetry assumption on W, one has

dW(x)F((gu)(x))𝑑x=dW(x)F(u(g-1x))𝑑x=dW(y)F(u(y))𝑑y

for every gO(d), the functional Jλ is O(d)-invariant on EV.

So, owing to Theorem 4, uλ is a weak solution of problem (Slambda). In this setting, in order to prove that uλ0 in FixO(d)(EV), first we claim that there exists a sequence of functions {wj}j in FixO(d)(EV) such that

lim supj+Ψ|FixO(d)(EV)(wj)Φ(wj)=+.(3.9)

By the assumption on the limsup in (3.1) there exists a sequence {tj}j(0,+) such that tj0+ as j+ and

limj+F(tj)tj2=+,

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

F(tj)>Mtj2.(3.10)

Now, following Kristály, Moroşanu, and O’Regan in [29], we construct a special test function belonging to FixO(d)(EV) that will be useful for our purposes. If a<b, define

Aab:={xd:a|x|b}.

Since WL(d){0} is a radially symmetric function with W0, one can find two real numbers R>r>0 and α>0 such that

essinfxArRW(x)α>0.(3.11)

Hence, let 0<r<R such that (3.11) holds and σ(0,12(R-r)). Set vσFixO(d)(EV) given by

vσ(x):={(|x|-rσ)+if |x|r+σ,1if r+σ|x|R-σ,(R-|x|σ)+if |x|R-σ,

where z+:=max{0,z}. With the above notation, we have:

  • (1)

    supp(vσ)ArR,

  • (2)

    vσ1,

  • (3)

    vσ(x)=1 for every xAr+σR-σ.

Now, define wj:=tjvσ for any j. Since vσFixO(d)(EV), of course one has wjFixO(d)(EV) for any j. Furthermore, taking into account the algebraic properties of the functions vσ stated in 13, since F(0)=0, and by using (3.10) we can write

Ψ|FixO(d)(EV)(wj)Φ(wj)=Ar+σR-σW(x)F(wj(x))𝑑x+ArRAr+σR-σW(x)F(wj(x))𝑑xΦ(wj)=Ar+σR-σW(x)F(tj)𝑑x+ArRAr+σR-σW(x)F(tjvσ(x))𝑑xΦ(wj)2M|Ar+σR-σ|αtj2+ArRAr+σR-σW(x)F(tjvσ(x))𝑑xtj2vσEV2(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 + and the one in which the liminf in (3.1) is finite.

Case 1. Suppose that limt0+F(t)t2=+. Then there exists ρM>0 such that for any t with 0<t<ρM,

F(t)Mt2.(3.13)

Since tj0+ and 0vσ(x)1 in d, it follows that wj(x)=tjvσ(x)0+ as j+ uniformly in xd. Hence, 0wj(x)<ρM for j sufficiently large and for any xd. Hence, as a consequence of (3.12) and (3.13), we have

Ψ|FixO(d)(EV)(wj)Φ(wj)2M|Ar+σR-σ|αtj2+ArRAr+σR-σW(x)F(tjvσ(x))𝑑xtj2vσEV22Mα|Ar+σR-σ|+ArRAr+σR-σ|vσ(x)|2𝑑xvσEV2

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

Case 2. Suppose that lim inft0+F(t)t2=. Then, for any ε>0, there exists ρε>0 such that for any t with 0<t<ρε,

F(t)(-ε)t2.(3.14)

Arguing as above, we can suppose that 0wj(x)=tjvσ(x)<ρε for j large enough and any xd. Thus, by (3.12) and (3.14) we get

Ψ|FixO(d)(EV)(wj)Φ(wj)2M|Ar+σR-σ|αtj2+ArRAr+σR-σW(x)F(tjvσ(x))𝑑xtj2vσEV22αM|Ar+σR-σ|+(-ε)ArRAr+σR-σ|vσ(x)|2𝑑xvσEV2,(3.15)

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

M>max{0,-2|Ar+σR-σ|ArRAr+σR-σ|vσ(x)|2𝑑x},

and ε>0 small enough so that

εArRAr+σR-σ|vσ(x)|2𝑑x<M2|Ar+σR-σ|+ArRAr+σR-σ|vσ(x)|2𝑑x,

by (3.15) we get

Ψ|FixO(d)(EV)(wj)Φ(wj)2αM|Ar+σR-σ|+(-ε)ArRAr+σR-σ|vσ(x)|2𝑑xvσEV22αvσEV2(M|Ar+σR-σ|+ArRAr+σR-σ|vσ(x)|2𝑑x-εArRAr+σR-σ|vσ(x)|2𝑑x)=αM|Ar+σR-σ|vσEV2

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

wjEV=tjvσEV0

as j+, so that for j large enough,

wjEV<2γ¯.

Thus

wjΦ-1((-,γ¯2)),(3.16)

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

𝒥λ(wj)=Φ(wj)-λΨ|FixO(d)(EV)(wj)<0(3.17)

for j sufficiently large. Since uλ is a global minimum of the restriction of 𝒥λ to Φ-1((-,γ¯2)) (see (3.8)), by (3.16) and (3.17) we conclude that

𝒥λ(uλ)𝒥λ(wj)<0=𝒥λ(0),(3.18)

so that uλ0 in FixO(d)(EV). Thus, uλ is a non-trivial weak solution of problem (Slambda). The arbitrariness of λ gives that uλ0 for any λ(0,λ).

Now, we claim that limλ0+uλEV=0. For this, let us fix λ(0,λ(γ¯)) for some γ¯>0. By Φ(uλ)<γ¯2 one has

Φ(uλ)=12uλEV2<γ¯2,

that is,

uλEV<2γ¯.

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

|dW(x)f(uλ(x))uλ(x)𝑑x|αf(dW(x)|uλ(x)|𝑑x+dW(x)|uλ(x)|q𝑑x)αf(Wpuλq+Wuλqq)<cqαf(2Wpγ¯+2q2cqq-1Wγ¯q)=:Mγ¯.(3.19)

Since uλ is a critical point of 𝒥λ, it follows that 𝒥λ(uλ),φ=0 for any φFixO(d)(EV) and every λ(0,λ(γ¯)). In particular, 𝒥λ(uλ),uλ=0, that is,

Φ(uλ),uλ=λdW(x)f(uλ(x))uλ(x)𝑑x(3.20)

for every λ(0,λ(γ¯)). Then, from (3.19) and (3.20), it follows that

0uλEV2=Φ(uλ),uλ=λdW(x)f(uλ(x))uλ(x)𝑑x<λMγ¯

for any λ(0,λ(γ¯)). We get limλ0+uλEV=0, as claimed.

Finally, a Strauss-type estimate (see [31] for details) proves that the functions uEV are homoclinic. Hence, the solution uλFixO(d)(EV){0}EV 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

limt0+f(t)t=+,

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

lim inft0+F(t)t2>-

is a technicality that we request in our proof in order to show that there exists a sequence {wj}jFixO(d)(EV) such that

lim supj+Ψ|FixO(d)(EV)(wj)Φ(wj)=+.

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

lim supt0+F(t)t2=+.

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

{-Δu+V(x)u=λf(x,u)in d,uH1(d),

where f:d× is a continuous function that satisfies:

  • (f1)

    There exists q(2,2*) such that

    supz|f(x,z)|1+|z|q-1W(x)a.e. in d

    for some radially symmetric function WL(Rd)Lqq-1(Rd){0} .

  • (f2)

    For every xRd and tR it follows that

    F(x,t)=F(gx,t)

    for every gO(d) .

  • (f3)

    For some x0Rd, there exists ϱ0>0 such that

    lim supt0+infB(x0,ϱ0)F(x,t)t2=+𝑎𝑛𝑑lim inft0+infB(x0,ϱ0)F(x,t)t2>-.

Here F(x,t):=0tf(x,z)𝑑z and B(x0,ϱ0) is the closed ball centered in x0 and radius ϱ0. See [27, 26] for related topics.

Remark 9.

We perform now the behavior of the functional 𝒥λ depending on the real parameter λ. In particular, we point out that by (3.18) the map

(0,λ)λ𝒥λ(uλ) is negative.(3.21)

Moreover, fixing γ¯>0, the function λ𝒥λ(uλ) is strictly decreasing in (0,λ(γ¯)), where

λ(γ¯):=qαfcq(γ¯q2Wp+2q2cqq-1Wγ¯q-1).

Indeed, let us write

𝒥λ(u)=λ(Φ(u)λ-Ψ|FixO(d)(EV)(u))(3.22)

for every uFixO(d)(EV). Now, fix 0<λ1<λ2<λ(γ¯) and let uλi be the global minimum of the functional 𝒥λi restricted to the sublevel Φ-1((-,γ¯2)) for i=1,2. We observe that

(Φ(uλi)λi-Ψ|FixO(d)(EV)(uλi))=infvΦ-1((-,γ¯2))(Φ(v)λi-Ψ|FixO(d)(EV)(v))

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

infvΦ-1((-,γ¯2))(Φ(v)λi-Ψ|FixO(d)(EV)(v))<0for i=1,2.

Moreover, it is easy to note that

infvΦ-1((-,γ¯2))(Φ(v)λ2-Ψ|FixO(d)(EV)(v))infvΦ-1((-,γ¯2))(Φ(v)λ1-Ψ|FixO(d)(EV)(v))(3.23)

owing to 0<λ1<λ2. Then, by (3.22)–(3.23) and again by the fact that 0<λ1<λ2, we have

𝒥λ2(uλ2)=λ2infvΦ-1((-,γ¯)2)(Φ(v)λ2-Ψ|FixO(d)(EV)(v))λ2infvΦ-1((-,γ¯2))(Φ(v)λ1-Ψ|FixO(d)(EV)(v))<λ1infvΦ-1((-,γ¯2))(Φ(v)λ1-Ψ|FixO(d)(EV)(v))=Jλ1(uλ1),

so that the real function λ𝒥λ(uλ) is (strictly) decreasing in (0,λ(γ¯)).

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 d6 and consider the subgroup Hd,iO(d) given by

Hd,i:={O(d2)×O(d2)if i=d-22,O(i+1)×O(d-2i-2)×O(i+1)if id-22,

for every iJ:={1,,τd}, where

τd:=(-1)d+[d-32].

Let us define the involution ηHd,i:dd as follows:

ηHd,i(x):={(x3,x1)if i=d-22, and x:=(x1,x3)d2×d2,(x3,x2,x1)if id-22, and x:=(x1,x2,x3)i+1×d-2i-2×i+1,

for every iJ. By definition, one has ηHd,iHd,i, as well as

ηHd,iHd,iηHd,i-1=Hd,iandηHd,i2=idd

for every iJ.

Moreover, for every iJ, let us consider the compact group

Hd,ηi:=Hd,i,ηHd,i,

that is, Hd,ηi=Hd,iηHd,iHd,i, and the action :Hd,ηi×EVEV of Hd,ηi on EV is given by

hu(x):={u(h-1x)if hHd,i,-u(g-1ηHd,i-1x)if h=ηHd,igHd,ηiHd,i,(4.1)

for every xd. We notice that is defined for every element of Hd,ηi. Indeed, if hHd,ηi, then either hHd,i or h=τgHd,ηiHd,i, with gHd,i. Moreover, set

FixHd,ηi(EV):={uEV:hu=u for all hHd,ηi}

for every iJ. Following Bartsch and Willem [13], for every iJ, the embedding

FixHd,ηi(EV)Lq(d)

is compact, for every q(2,2*).

Finally, the following facts hold:

  • If d=4 or d6, then

    FixHd,ηi(EV)FixO(d)(EV)={0}(4.2)

    for every iJ.

  • If d=6 or d8, then

    FixHd,ηi(EV)FixHd,ηj(EV)={0}(4.3)

    for every i,jJ and ij.

See [29, Theorem 2.2] for details.

Remark 10.

We notice that, if we consider the elliptic problem

{-Δu+VHd,ηi(x)u=λWHd,ηi(x)f(u)in d,uH1(d),

requiring that VHd,ηi and WHd,ηi be Hd,ηi-invariant (instead of radially symmetric) under the action of the group Hd,ηi on d, for some iJ, then Theorem 6 ensures the existence of at least one non-trivial solution.

Let

ci,q:=sup{uquEV:uFixHd,ηi(EV){0}}

for every q(2,2*), and iJ. Setting

λi,q:=qαfci,qmaxγ>0(γq2Wp+2q2ci,qq-1Wγq-1),(4.4)

our main result reads as follows.

Theorem 11.

Assume d>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 λ given by

λ:={λif d=5,min{λ,λi,q:iJ}if d5,

such that, for every λ(0,λ), the problem

{-Δu+V(x)u=λW(x)f(u)in d,uH1(d),(Slambda’)

admits at least

ζS(d):=1+(-1)d+[d-32]

pairs of non-trivial weak solutions {±uλ,i}iJEV, where J:={1,,ζS(d)}, such that

limλ0+uλ,iEV=0,

and |uλ,i(x)|0, as |x|, for every iJ. Moreover, if d5, problem (Slambda’) admits at least

τd:=(-1)d+[d-32]

pairs of sign-changing weak solutions {±zλ,i}iJEV.

Proof.

We divide the proof into two parts.

Part 1: Dimension d=5. Since f is odd, the energy functional

𝒥λ(u):=Φ(u)-λΨ|FixO(d)(EV)(u)for all uFixO(d)(EV)

is even. Owing to Theorem 6, for every λ(0,λ), problem (Slambda’) admits at least one (that is, ζS(5)=1) non-trivial pair of radial weak solutions {±uλ}EV. Furthermore, the functions ±uλ are homoclinic and

limλ0+uλEV=0.

This concludes the first part of the proof.

Part 2: Dimension d>3 and d5. For every λ>0 and i=1,2,,τd, consider the restrictions

λ,i:=Jλ|FixHd,ηi(EV):FixHd,ηi(EV)

defined by

λ,i:=ΦHd,ηi(u)-λΨ|FixHd,ηi(EV)(u),

where

ΦHd,ηi(u):=12uEV2andΨ|FixHd,ηi(EV)(u):=dW(x)F(u(x))𝑑x

for every uFixHd,ηi(EV). In order to obtain the existence of

τd:=(-1)d+[d-32]

pairs of sign-changing weak solutions {±zλ,i}iJEV, where J:={1,,τd}, the main idea of the proof consists in applying Theorem 3 to the functionals λ,i for every iJ. We notice that, since d>3 and d5, τd1. Consequently, the cardinality |J|1.

Since 0<λ<λi,q, with iJ, there exists γ¯i>0 such that

λ<λ(i)(γ¯i):=qαfci,q(γ¯iq2Wp+2q2ci,qq-1Wγ¯iq-1).

Similar arguments used proving (3.7) yield

φ(γ¯i2)χ(γ¯i2)αfcq(2Wpγ¯i+2q2cqq-1qWγ¯iq-2)<1λ.

Thus,

λ(0,qαfcq(γ¯q2Wp+2q2cqq-1Wγ¯q-1))(0,1φ(γ¯i2)).

Thanks to Theorem 3, there exists a function zλ,iΦHd,ηi-1((-,γ¯i2)) such that

ΦHd,ηi(zλ,i)-λ(Ψ|FixHd,ηi(EV))(zλ,i)=0,

and, in particular, zλ,i is a global minimum of the restriction of λ,i to ΦHd,ηi-1((-,γ¯i2)).

Due to the evenness of Jλ, bearing in mind (4.1), and thanks to the symmetry assumptions on the potentials V and W, we have that Jλ(hu)=Jλ(u) for every hHd,ηi and uEV, i.e., the functional Jλ is Hd,ηi-invariant on EV. Indeed, Hd,ηi acts isometrically on EV (note that V is radial) and, thanks to the symmetry assumption on W, one has

dW(x)F(hu(x))𝑑x=dW(x)F(u(h-1x))𝑑x=dW(y)F(u(y))𝑑y

if hHd,i, and

dW(x)F(hu(x))𝑑x=dW(x)F(u(g-1ηHd,i-1x))𝑑x=dW(y)F(u(y))𝑑y

if h=ηHd,igHd,ηiHd,i.

On account of the principle of symmetric criticality (recalled in Theorem 4), the critical point pairs {±zλ,i} of λ,i are also critical points of Jλ. Now, we have to show that the solution zλ,i found here above is not the trivial function. If f(0)0, then it easily follows that zλ,i0 in FixHd,ηi(EV), since the trivial function does not solve problem (Slambda’). Let us consider the case when f(0)=0 and let zλ,i be such that

λ,i(zλ,i)λ,i(u)for any uFixHd,ηi(EV) such that ΦHd,ηi(u)<γ¯i2

and

ΦHd,ηi(zλ,i)<γ¯i2,

and also zλ,i is a critical point of λ,i in FixHd,ηi(EV). In this setting, in order to prove that we have zλ,i0 in FixHd,ηi(EV), first we claim that there exists a sequence {wji}j in FixHd,ηi(EV) such that

lim supj+Ψ|FixHd,ηi(EV)(wji)Φ(wji)=+.(4.5)

In order to construct the sequence {wji}jFixHd,ηi(EV) 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<r<R be such that condition (3.11) holds and rR5+42. Set σ(0,1) and define vσiEV as follows:

vσi(x):={vσd-22(x)if i=d-22, and x:=(x1,x3)d2×d2,viσ(x)if id-22, and x:=(x1,x2,x3)i+1×d-2i-2×i+1,

for every xd, where

vσd-22(x1,x3):=[(R-r4-max{(|x1|2-R+3r4)2+|x3|2,σR-r4})+-(R-r4-max{(|x1|2-R+3r4)2+|x3|2,σR-r4})+]×4(R-r)(1-σ)

for every (x1,x3)d2×d2 and

viσ(x1,x2,x3):=[(R-r4-max{(|x1|2-R+3r4)2+|x3|2,σR-r4})+-(R-r4-max{(|x3|2-R+3r4)2+|x1|2,σR-r4})+]×(R-r4-max{|x2|,σR-r4})+4(R-r)2(1-σ)2

for every (x1,x2,x3)d2×d-2i-2×d2, and id-22. Now, it is possible to prove that vσiFixHd,ηi(EV). Moreover, for every ϱ(0,1], let

Qϱ(1):={(x1,x3)i+1×i+1:(|x1|2-R+3r4)2+|x3|2ϱR-r4},Qϱ(2):={(x1,x3)i+1×i+1:(|x3|2-R+3r4)2+|x1|2ϱR-r4}.

Define

Dϱi:={Dϱd-22if i=d-22,Diϱif id-22,

where

Dϱd-22:={(x1,x3)d2×d2:(x1,x3)Qϱ(1)Qϱ(2)},Diϱ:={(x1,x2,x3)d2×d-2i-2×d2:(x1,x3)Qϱ(1)Qϱ(2) and |x2|ϱR-r4}

for every id-22. The sets Dϱi have positive Lebesgue measure and they are Hd,ηi-invariant. Moreover, for every σ(0,1), one has vσiFixHd,ηi(EV) and the following facts hold:

  • (1)

    supp(vσi)=D1iA[r,R],

  • (2)

    vσi1,

  • (3)

    |vσi(x)|=1 for every xDσi.

Thus, let wji:=tjvσi for any j. Of course, wjiFixHd,ηi(EV) for any j. Furthermore, taking into account the properties of vσi stated in 13, since F is even (this implies that F(wji(x))=F(tj) for every xDσi) with F(0)=0, and by using (3.10) one has for j sufficiently large,

Ψ|FixHd,ηi(EV)(wji)Φ(wji)=DσiW(x)F(wji(x))𝑑x+ArRDσiW(x)F(wji(x))𝑑xΦ(wji)=DσiW(x)F(tj)𝑑x+ArRDσiW(x)F(tjvσi(x))𝑑xΦ(wji)2M|Dσi|αtj2+ArRDσiW(x)F(tjvσi(x))𝑑xtj2vσiEV2.(4.6)

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

λ,i(zλ,i)λ,i(wji)<0=λ,i(0),

so that zλ,i0 in FixHd,ηi(EV). In addition, by adapting again the arguments used along the proof of Theorem 6 it follows that

limλ0+zλ,iEV=0,

and |zλ,i(x)|0 as |x|.

On the other hand, since λ<λ 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 {±uλ}EV. Moreover,

limλ0+uλEV=0,

and |uλ(x)|0 as |x|.

In conclusion, since λ<λ, there exist τd+1 positive numbers γ¯, γ¯1,,γ¯τd such that

±uλΦ-1((-,γ¯2)){0}FixO(d)(EV)and±zλ,iΦHd,ηi-1((-,γ¯22)){0}FixHd,ηi(EV).

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

Φ-1((-,γ¯2))ΦHd,ηi-1((-,γ¯i2)){0}=

for every iJ and

ΦHd,ηi-1((-,γ¯i2))ΦHd,ηj-1((-,γ¯j2)){0}=

for every i,jJ and ij. Consequently, problem (Slambda’) admits at least

ζS(d):=τd+1

pairs of non-trivial weak solutions {±uλ,i}iJEV, where J:={1,,ζS(d)}, such that

limλ0+uλ,iEV=0,

and |uλ,i(x)|0, as |x|, for every iJ. Moreover, by construction, it follows that

τd:=(-1)d+[d-32]

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 λ(0,λ).

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<r<2 and 2<s<4 and let V be a potential for which (hV4) holds. Then, owing to Theorem 11, for λ sufficiently small, the problem

{-Δu+V(x)u=λ(1+|x|4)2(s-1)s(|u|r-2u+|u|s-2u)in 4,uH1(4),(Plambda)

admits at least one non-trivial pair of radially symmetric weak solutions {±uλ} and one pair of sign-changing weak solutions {±zλ}. Moreover,

limλ0+uλEV=limλ0+zλEV=0,

and |uλ(x)|0, as well as |zλ(x)|0, as |x|. More precisely, set

Ws:=(41(1+|x|4)2𝑑x)s-1s=(area(𝕊3)4)s-1s,

where 𝕊3 denotes the 4-dimensional Euclidean unit sphere and

κs:=sup{uquEV:uFixH4,τ(EV){0}},

where τ:44 is the involution given by τ(x1,x3):=(x3,x1) for every (x1,x3)2×2 and associated to the subgroup O(2)×O(2)O(4). If

cs:=sup{usuEV:uFixO(4)(EV){0}},

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

0<λ<s2min{1κsmaxγ>0(γs2Ws+2s2κss-1γs-1),1csmaxγ>0(γs2Ws+2s2css-1γs-1)}.

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(t)=o(|t|) as |t|,

  • (h2)

    f(t)=o(|t|) as |t|0,

  • (h3)

    there exists t0 such that F(t0)>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

limt0+f(t)t=limt0+tr-1+ts-1t=+,

and consequently (h2) 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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About the article


Received: 2018-03-16

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, DOI: https://doi.org/10.1515/acv-2018-0016.

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