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Volume 8, Issue 1

# Existence of a bound state solution for quasilinear Schrödinger equations

Yan-Fang Xue
• School of Mathematics and Statistics, Southwest University, Chongqing 400715; and College of Mathematics and Information Sciences, Xin-Yang Normal University, Xinyang, Henan 464000, P. R. China
• Email
• Other articles by this author:
/ Chun-Lei Tang
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/anona-2016-0244

## Abstract

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ${ℝ}^{N}$. After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in ${H}^{1}\left({ℝ}^{N}\right)$. The proofs are based on the Pohozaev manifold and a linking theorem.

MSC 2010: 35J62; 35J20; 35B09

## 1 Introduction and main result

In the present paper, we consider a class of quasilinear Schrödinger equations of the form

$i\frac{\partial \psi }{\partial t}=-\mathrm{△}\psi +W\left(x\right)\psi -l\left({|\psi |}^{2}\right)\psi -\kappa \left[\mathrm{△}\rho \left({|\psi |}^{2}\right)\right]{\rho }^{\prime }\left({|\psi |}^{2}\right)\psi ,$(1.1)

where $W:{ℝ}^{N}\to ℝ$ is a given potential, $N\ge 3$, κ is a positive constant, and l, ρ are real functions. Corresponding to various types of nonlinear terms ρ, this problem appears naturally in different mathematical physical models; see [28, 23, 21] for an explanation. Here we focus on the case $\rho \left(s\right)=s$, $\kappa =1$. As we all know, a standing wave of (1.1) is a solution of the form $\psi \left(t,x\right)=\mathrm{exp}\left(-iEt\right)u\left(x\right)$, $E\in ℝ$, and consequently $u\left(x\right)$ satisfies the following equation:

$-\mathrm{\Delta }u+V\left(x\right)u-\mathrm{\Delta }\left({u}^{2}\right)u=g\left(u\right),x\in {ℝ}^{N},N\ge 3,$(1.2)

where $V\left(x\right)=W\left(x\right)-E$ is the new potential and $g\left(u\right)=l\left({u}^{2}\right)u$ is the new nonlinear term. The main purpose of this paper is to deal with the existence of solutions for equation (1.2). This kind of problem has been studied by many authors; see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31, 24, 32, 27, 9, 7, 25, 13, 8, 30] and the references therein.

In [28, 23], by using a constrained minimization argument, a positive ground state solution has been proved for equation (1.2) with $g\left(u\right)=\lambda {|u|}^{q-1}u$, $4\le q+1<2\cdot {2}^{*}$, where ${2}^{*}=2N/\left(N-2\right)$ is the Sobolev critical exponent. The case $4\le q+1<2\cdot {2}^{*}$ is called subcritical growth; there are many articles that deal with this class of problem (see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31]). In [22], the existence of both positive and sign-changing ground states of soliton-type solutions were established via the Nehari method. Then by a change of variables, the quasilinear problem is transformed into a semilinear one; see [21] for an Orlicz space framework and [6] for a Sobolev space frame. Recently, a perturbation method was developed in [25] to deal with equation (1.2), which can be applied to more general quasilinear Schrödinger equations (see also [20]).

For the critical case, we would like to mention [32, 27, 9, 26, 7, 25, 13, 8, 30] and the references therein. It seems that Moameni [27] first studied the critical case when the potential V is radial and satisfies some geometric conditions. Do Ó, Miyagaki and Soares [9] obtained a positive classical solution by using the concentration compactness principle of Lions [19]. He and Li [13] obtained the existence, concentration and multiplicity of weak solutions by employing the minimax theorems and Ljusternik–Schnirelmann theory.

In recent years, the use of the Pohozaev manifold was shown very effective when treading nonlinearities which do not satisfy the Ambrosetti–Rabinowitz condition and the monotonicity condition; see [16, 5, 14, 15, 4]. Lehrer and Maia [16] employed the minimization methods restricted to the Pohozaev manifold to obtain the existence of positive solutions for the asymptotically linear case. Later in [5], Carrião, Lehrer and Miyagaki extended the result given in [16] to more general quasilinear equations. Motivated by [16, 5, 17], we consider equation (1.2) with the nonlinear term $g\left(u\right)$ being nonhomogeneous and asymptotically linear at infinity. We will use the linking theorem together with the barycenter function restricted to the Pohozaev manifold associated to our problem.

The main obstacle in finding a solution of equation (1.2) is due to the influence of the quasilinear and nonconvex term $\mathrm{\Delta }\left({u}^{2}\right)u$. The other difficulty is the possible lack of compactness due to the unboundedness of the domain. We will employ an argument developed in [6] to overcome the first difficult and a splitting lemma to conquer the second one.

We suppose that V satisfies the following assumptions:

• (V1)

$V\in {C}^{2}\left({ℝ}^{N},ℝ\right)$;

• (V2)

${lim}_{|x|\to \mathrm{\infty }}V\left(x\right)={V}_{\mathrm{\infty }}<1$, $V\left(x\right)>{V}_{\mathrm{\infty }}>0$ for all $x\in {ℝ}^{N}$;

• (V3)

$\left(\nabla V\left(x\right),x\right)\le 0$ for all $x\in {ℝ}^{N}$, and the strict inequality holds on a subset of positive Lebesgue measure of ${ℝ}^{N}$;

• (V4)

$NV\left(x\right)+\left(\nabla V\left(x\right),x\right)\ge N{V}_{\mathrm{\infty }}$ for all $x\in {ℝ}^{N}$;

• (V5)

$\frac{x{H}_{V}\left(x\right)x}{N}+\left(\nabla V\left(x\right),x\right)\le 0$ for all $x\in {ℝ}^{N}$, where ${H}_{V}$ is the Hessian matrix of the function V.

We assume the following conditions on the function g:

• (g1)

$g\in {C}^{1}\left({ℝ}^{+},{ℝ}^{+}\right)$ and ${lim}_{s\to {0}^{+}}\frac{g\left(s\right)}{s}=0$;

• (g2)

${lim}_{s\to \mathrm{\infty }}\frac{g\left(s\right)}{{s}^{2}}=1$;

• (g3)

$Q\left(s\right):=\frac{1}{4}g\left(s\right)s-G\left(s\right)>0$ and ${lim}_{s\to \mathrm{\infty }}Q\left(s\right)=+\mathrm{\infty }$, where $G\left(s\right)={\int }_{0}^{s}g\left(t\right)𝑑t$.

We employ an argument developed in [6] to introduce a variational framework associated with equation (1.2). We observe that equation (1.2) is formally the Euler–Lagrange equation associated with the energy functional

$J\left(u\right)=\frac{1}{2}{\int }_{{ℝ}^{N}}\left[\left(1+2{u}^{2}\right){|\nabla u|}^{2}\right]𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){u}^{2}𝑑x-{\int }_{{ℝ}^{N}}G\left(u\right)𝑑x.$

We make a change of variables $v:={f}^{-1}\left(u\right)$, where f is defined by

${f}^{\prime }\left(t\right)=\frac{1}{{\left(1+2{f}^{2}\left(t\right)\right)}^{1/2}},$$\mathrm{ }t\in \left[0,+\mathrm{\infty }\right),$$f\left(t\right)=-f\left(-t\right),$$\mathrm{ }t\in \left(-\mathrm{\infty },0\right].$

After the change of variables from J, we obtain the following functional:

$I\left(v\right)=\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x-{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$

Then $I\left(v\right)=J\left(u\right)=J\left(f\left(v\right)\right)$ and I is well defined on ${H}^{1}\left({ℝ}^{N}\right)$, $I\in {C}^{1}\left({H}^{1}\left({ℝ}^{N}\right),ℝ\right)$ under the hypotheses (V${}_{1}$), (V${}_{2}$) and (g${}_{1}$)–(g${}_{3}$). Moreover, we observe that if v is a critical point of the functional I, then the function $u=f\left(v\right)$ is a solution of equation (1.2) (see [6]).

The critical points of I are weak solutions of the problem

$-\mathrm{\Delta }v+V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)=g\left(f\left(v\right)\right){f}^{\prime }\left(v\right).$(1.3)

We can demonstrate that

$〈{I}^{\prime }\left(v\right),\phi 〉={\int }_{{ℝ}^{N}}\nabla v\cdot \nabla \phi dx+{\int }_{{ℝ}^{N}}V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)\phi 𝑑x-{\int }_{{ℝ}^{N}}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)\phi 𝑑x$

for all $v,\phi \in {H}^{1}\left({ℝ}^{N}\right)$. Each solution of equation (1.3) satisfies $\gamma \left(v\right)=0$, where

$\gamma \left(v\right)=\frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x-N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$

We define the Pohozaev manifold associated with equation (1.3) by

$\mathcal{𝒫}:=\left\{v\in {H}^{1}\left({ℝ}^{N}\right):v\ne 0,\gamma \left(v\right)=0\right\}.$

Let c be the min-max mountain pass level for the functional I given by

$c=\underset{\zeta \in \mathrm{\Gamma }}{inf}\underset{t\in \left[0,1\right]}{\mathrm{max}}I\left(\zeta \left(t\right)\right),$

where

$\mathrm{\Gamma }=\left\{\zeta \in C\left(\left[0,1\right],{H}^{1}\left({ℝ}^{N}\right)\right):\zeta \left(0\right)=0\ne \zeta \left(1\right),I\left(\zeta \left(1\right)\right)<0\right\}.$

Under the previous hypotheses on V and g, we have the following nonexistence result.

#### Theorem 1.1.

Suppose that (V${}_{1}$)(V${}_{5}$) and (g${}_{1}$)(g${}_{3}$) are satisfied. Then $p\mathrm{:=}{\mathrm{inf}}_{v\mathrm{\in }\mathcal{P}}\mathit{}I\mathit{}\mathrm{\left(}v\mathrm{\right)}$ is not a critical level for the functional I. In particular, the infimum p is not achieved.

Consider now also the limiting problem

$-\mathrm{\Delta }v+{V}_{\mathrm{\infty }}f\left(v\right){f}^{\prime }\left(v\right)=g\left(f\left(v\right)\right){f}^{\prime }\left(v\right).$(1.4)

Its associated energy functional is denoted by ${I}_{\mathrm{\infty }}$.

${I}_{\mathrm{\infty }}\left(v\right)=\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}{V}_{\mathrm{\infty }}{f}^{2}\left(v\right)𝑑x-{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$

Each solution of equation (1.4) satisfies the following Pohozaev identity:

$\frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N}{2}{\int }_{{ℝ}^{N}}{V}_{\mathrm{\infty }}{f}^{2}\left(v\right)𝑑x=N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$(1.5)

We define the Pohozaev manifold associated with equation (1.4) by

and ${p}_{\mathrm{\infty }}:={inf}_{v\in {\mathcal{𝒫}}_{\mathrm{\infty }}}{I}_{\mathrm{\infty }}\left(v\right)$.

We say that a solution $v\in {H}^{1}\left({ℝ}^{N}\right)$ of equation (1.4) is a least energy solution if and only if ${I}_{\mathrm{\infty }}\left(v\right)={m}_{\mathrm{\infty }}$, where

We define

${\mathrm{\Gamma }}_{\mathrm{\infty }}=\left\{\zeta \in C\left(\left[0,1\right],{H}^{1}\left({ℝ}^{N}\right)\right):\zeta \left(0\right)=0\ne \zeta \left(1\right),{I}_{\mathrm{\infty }}\left(\zeta \left(1\right)\right)<0\right\},$

as well as the mountain pass min-max level

${c}_{\mathrm{\infty }}=\underset{\zeta \in \mathrm{\Gamma }}{inf}\underset{t\in \left[0,1\right]}{\mathrm{max}}{I}_{\mathrm{\infty }}\left(\zeta \left(t\right)\right).$

Using a method similar to the one in [15], we can deduce that ${c}_{\mathrm{\infty }}={m}_{\mathrm{\infty }}={p}_{\mathrm{\infty }}$.

Now we can state our main existence result.

#### Theorem 1.2.

Suppose that (V${}_{1}$)(V${}_{5}$) and (g${}_{1}$)(g${}_{3}$) are satisfied and the following facts hold:

• (1)

$g\in \mathrm{Lip}\left({ℝ}^{+},{ℝ}^{+}\right)$ ;

• (2)

${\parallel V\left(x\right)-{V}_{\mathrm{\infty }}\parallel }_{\mathrm{\infty }}$ is sufficiently small;

• (3)

the least energy level ${c}_{\mathrm{\infty }}$ of equation ( 1.4 ) is an isolated radial critical level for ${I}_{\mathrm{\infty }}$ or equation ( 1.4 ) admits a unique positive solution which is radially symmetric about some point.

Then equation (1.2) admits a positive solution whose energy is above ${c}_{\mathrm{\infty }}$.

#### Remark 1.3.

These results extend the corresponding results in [17] to the more general quasilinear case. The framework employed and ideas of the proofs for our main results are close to those found in [17]. However, some technical details in this paper are different from those in [17].

#### Remark 1.4.

For the case of nonlinearities $g\left(u\right)={|u|}^{p-1}u$, uniqueness properties of ground state solutions of equation (1.2) were recently proved in [1, 12, 29, 2], so assumption of Theorem 1.2 (3) is expected to be fulfilled.

#### Example 1.5.

Let $V\left(x\right)=a+b{\left(1+{|x|}^{2}\right)}^{-c}$, where $0, $0 and $b>0$ is small enough. Then

$\left(\nabla V\left(x\right),x\right)=-2bc{|x|}^{2}{\left(1+{|x|}^{2}\right)}^{-c-1}$

and

$x{H}_{V}\left(x\right)x=2bc{\left(1+{|x|}^{2}\right)}^{-c-2}\left[2\left(c+1\right){|x|}^{4}-\left(1+{|x|}^{2}\right){|x|}^{2}\right].$

We can see by a computation that $V\left(x\right)$ satisfies all conditions in Theorem 1.2.

#### Remark 1.6.

Conditions (g${}_{1}$) and (g${}_{2}$) imply that given $\epsilon >0$ and $3\le q\le 2\cdot {2}^{*}$, there exists a positive constant ${C}_{\epsilon }=C\left(\epsilon ,q\right)$ such that

(1.6)

We also obtain the estimate

#### Remark 1.7.

Since we are looking for positive solutions, we set $g\left(s\right)=0$ for all $s<0$. Let v be a critical point of I. Taking $\phi =-{v}^{-}$, we have

${\int }_{{ℝ}^{N}}{|\nabla {v}^{-}|}^{2}𝑑x+{\int }_{{ℝ}^{N}}V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)\left(-{v}^{-}\right)𝑑x=0.$

Since $f\left(v\right)\left(-{v}^{-}\right)\ge 0$, we get

${\int }_{{ℝ}^{N}}{|\nabla {v}^{-}|}^{2}𝑑x=0$

and

${\int }_{{ℝ}^{N}}\frac{V\left(x\right)f\left(v\right)\left(-{v}^{-}\right)}{\sqrt{1+2{f}^{2}\left(v\right)}}𝑑x=0.$

Hence we may conclude that ${v}^{-}=0$ a.e. in ${ℝ}^{N}$ and $v={v}^{+}\ge 0$. As $u=f\left(v\right)$, we conclude that u is a nonnegative solution for equation (1.2).

#### Notation.

In this paper, we use the following notations:

• ${H}^{1}\left({ℝ}^{N}\right)$ is the usual Hilbert space endowed with the norm

${\parallel u\parallel }^{2}={\int }_{{ℝ}^{N}}\left({|\nabla u|}^{2}+{u}^{2}\right)𝑑x.$

• ${L}^{s}\left({ℝ}^{N}\right)$ is the usual Banach space endowed with the norm

• ${\parallel u\parallel }_{\mathrm{\infty }}=\mathrm{ess}{sup}_{x\in {ℝ}^{N}}|u\left(x\right)|$ denotes the usual norm in ${L}^{\mathrm{\infty }}\left({ℝ}^{N}\right)$.

• ${B}_{r}\left(y\right)=\left\{x\in {ℝ}^{N}:|x-y|, ${B}_{r}=\left\{x\in {ℝ}^{N}:|x|.

• ${u}^{+}=\mathrm{max}\left\{u,0\right\}$, ${u}^{-}=\mathrm{max}\left\{-u,0\right\}$.

• $|\mathrm{\Omega }|$ denotes the Lebesgue measure of the set Ω.

• $C,{C}_{\epsilon },{C}_{1},{C}_{2},\mathrm{\dots }$ denote various positive constants whose exact value is inessential.

## 2 Some preliminary results

In this section, we first summarize the properties of f, which have been proved in [6, 10].

#### Lemma 2.1.

The function f satisfies the following properties:

• (1)

f is uniquely defined, ${C}^{\mathrm{\infty }}$ and invertible;

• (2)

$|{f}^{\prime }\left(t\right)|\le 1$ for all $t\in ℝ$ ;

• (3)

$|f\left(t\right)|\le |t|$ for all $t\in ℝ$ ;

• (4)

$f\left(t\right)/t\to 1$ as $t\to 0$ ;

• (5)

$f\left(t\right)/\sqrt{t}\to {2}^{1/4}$ as $t\to \mathrm{\infty }$ ;

• (6)

$f\left(t\right)/2\le t{f}^{\prime }\left(t\right)\le f\left(t\right)$ for all $t>0$ ;

• (7)

$|f\left(t\right)|\le {2}^{1/4}{|t|}^{1/2}$ for all $t\in ℝ$ ;

• (8)

${f}^{\prime }\left(t\right)\to 1$ as $t\to 0$ ;

• (9)

there exists a positive constant C such that $|f\left(t\right)|\ge C|t|$ for $|t|\le 1$ and $|f\left(t\right)|\ge C{|t|}^{1/2}$ for $|t|\ge 1$ ;

• (10)

$|f\left(t\right){f}^{\prime }\left(t\right)|\le 1/\sqrt{2}$ for all $t\in ℝ$.

#### Lemma 2.2.

The functional γ and the Pohozaev manifold $\mathcal{P}$ satisfy the following properties:

• (1)

$\left\{v\equiv 0\right\}$ is an isolated point of ${\gamma }^{-1}\left(\left\{0\right\}\right)$ ;

• (2)

$\mathcal{𝒫}$ is a closed set;

• (3)

$\mathcal{𝒫}$ is a ${C}^{1}$ manifold;

• (4)

there is a $\sigma >0$ such that $\parallel v\parallel >\sigma$ for all $v\in \mathcal{𝒫}$.

#### Proof.

(1) Thanks to Lemma 2.1 (9), we can deduce that there is ${C}_{1}>0$ such that

${f}^{2}\left(t\right)\ge {C}_{1}\left({t}^{2}-{|t|}^{{2}^{*}}\right).$(2.1)

By (V${}_{4}$), (1.6), (2.1), Lemma 2.1 (3), (7), and the Sobolev inequality, we have

$\gamma \left(v\right)=\frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x-N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x$$\ge \frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}N{V}_{\mathrm{\infty }}{f}^{2}\left(v\right)𝑑x-N{\int }_{{ℝ}^{N}}\left(\frac{\epsilon }{2}{|f\left(v\right)|}^{2}+{C}_{\epsilon }{|f\left(v\right)|}^{2\cdot {2}^{*}}\right)𝑑x$$\ge \frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N{V}_{\mathrm{\infty }}}{2}{\int }_{{ℝ}^{N}}{C}_{1}\left({v}^{2}-{|v|}^{{2}^{*}}\right)𝑑x-N{\int }_{{ℝ}^{N}}\left(\frac{\epsilon }{2}{|v|}^{2}+{2}^{\frac{{2}^{*}}{2}}{C}_{\epsilon }{|v|}^{{2}^{*}}\right)𝑑x$$\ge \mathrm{min}\left\{\frac{N-2}{2},\frac{N{V}_{\mathrm{\infty }}{C}_{1}}{2}-\frac{\epsilon N}{2}\right\}{\parallel v\parallel }^{2}-\left(\frac{N{V}_{\mathrm{\infty }}{C}_{1}}{2}+{2}^{\frac{{2}^{*}}{2}}{C}_{\epsilon }N\right){S}^{-\frac{{2}^{*}}{2}}{\parallel v\parallel }^{{2}^{*}}$$={C}_{2}{\parallel v\parallel }^{2}-{C}_{3}{\parallel v\parallel }^{{2}^{*}},$

where S is the best Sobolev constant of the embedding ${\mathcal{𝒟}}^{1,2}\left({ℝ}^{N}\right)↪{L}^{{2}^{*}}\left({ℝ}^{N}\right)$ and

${C}_{2}=\mathrm{min}\left\{\frac{N-2}{2},\frac{N{V}_{\mathrm{\infty }}{C}_{1}}{2}-\frac{\epsilon N}{2}\right\}>0,{C}_{3}=\left(\frac{N{V}_{\mathrm{\infty }}{C}_{1}}{2}+{2}^{\frac{{2}^{*}}{2}}{C}_{\epsilon }N\right){S}^{-\frac{{2}^{*}}{2}}>0$

by taking $\epsilon >0$ sufficiently small. Let $0<\rho <1$ such that ${\rho }^{{2}^{*}}<{C}_{2}{\rho }^{2}/\left(2{C}_{3}\right)$; then if $\parallel v\parallel =\rho$, we have

$\gamma \left(v\right)\ge {C}_{2}{\rho }^{2}-{C}_{3}{\rho }^{{2}^{*}}>\frac{{C}_{2}}{2}{\rho }^{2}>0.$

(2) The functional $\gamma \left(v\right)$ is a ${C}^{1}$ functional, thus $\mathcal{𝒫}\cup \left\{0\right\}={\gamma }^{-1}\left(\left\{0\right\}\right)$ is a closed subset. Moreover, $\left\{v\equiv 0\right\}$ is an isolated point in ${\gamma }^{-1}\left(\left\{0\right\}\right)$ and the assertion follows.

(3) It follows from Lemma 2.1 (6) and (g${}_{3}$) that

$\frac{1}{2}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)v-G\left(f\left(v\right)\right)\ge \frac{1}{4}g\left(f\left(v\right)\right)f\left(v\right)-G\left(f\left(v\right)\right)>0.$(2.2)

Since $v\in \mathcal{𝒫}$, we have

$\frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x=N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$(2.3)

Hence we can deduce that

${\gamma }^{\prime }\left(v\right)v=\left(N-2\right){\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+N{\int }_{{ℝ}^{N}}V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)v𝑑x+{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right)f\left(v\right){f}^{\prime }\left(v\right)v𝑑x-N{\int }_{{ℝ}^{N}}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)v𝑑x.$$=2N{\int }_{{ℝ}^{N}}\left(G\left(f\left(v\right)\right)-\frac{1}{2}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)v\right)𝑑x+{\int }_{{ℝ}^{N}}\left[NV\left(x\right)+\left(\nabla V\left(x\right),x\right)\right]\left(f\left(v\right){f}^{\prime }\left(v\right)v-{f}^{2}\left(v\right)\right)𝑑x.$

Combining this with (2.2), Lemma 2.1 (6) and (V${}_{4}$), we have ${\gamma }^{\prime }\left(v\right)v<0$ if $v\in \mathcal{𝒫}$. This shows that $\mathcal{𝒫}$ is a ${C}^{1}$ manifold.

(4) Since 0 is an isolated point in ${\gamma }^{-1}\left(\left\{0\right\}\right)$, there must be a ball $\parallel v\parallel \le \sigma$ which does not intersect $\mathcal{𝒫}$ and the assertion is proved. ∎

#### Lemma 2.3.

Assume (V${}_{1}$), (V${}_{5}$) and (g${}_{1}$) hold. Then $\mathcal{P}$ is a nature constraint for the functional I.

#### Proof.

Let $v\in \mathcal{𝒫}$ be a critical point of ${I|}_{\mathcal{𝒫}}$. By the theorem of Lagrange multipliers, there exists a $\mu \in ℝ$ such that ${I}^{\prime }\left(v\right)+\mu {\gamma }^{\prime }\left(v\right)=0$. The proof is complete as soon as we show that $\mu =0$. Evaluating the linear functional above at $v\in \mathcal{𝒫}$, we obtain

${I}^{\prime }\left(v\right)v+\mu {\gamma }^{\prime }\left(v\right)v=0,$

namely

${\int }_{{ℝ}^{N}}{|\nabla v|}^{2}dx+{\int }_{{ℝ}^{N}}V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)vdx-{\int }_{{ℝ}^{N}}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)vdx+\mu \left[\left(N-2\right){\int }_{{ℝ}^{N}}{|\nabla v|}^{2}dx+N{\int }_{{ℝ}^{N}}V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)vdx$$+{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right)f\left(v\right){f}^{\prime }\left(v\right)vdx-N{\int }_{{ℝ}^{N}}g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)vdx\right]=0.$

This expression is associated with the equation

$-\mathrm{\Delta }v+V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)-g\left(f\left(v\right)\right){f}^{\prime }\left(v\right)+\mu \left[-\left(N-2\right)\mathrm{\Delta }v+NV\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)+\left(\nabla V\left(x\right),x\right)f\left(v\right){f}^{\prime }\left(v\right)-Ng\left(f\left(v\right)\right){f}^{\prime }\left(v\right)\right]=0,$

which can be rewritten as

$-\left[1+\mu \left(N-2\right)\right]\mathrm{\Delta }v+\left(1+\mu N\right)V\left(x\right)f\left(v\right){f}^{\prime }\left(v\right)+\mu \left(\nabla V\left(x\right),x\right)f\left(v\right){f}^{\prime }\left(v\right)=\left(1+N\mu \right)g\left(f\left(v\right)\right){f}^{\prime }\left(v\right).$(2.4)

Each solution of equation (2.4) satisfies $\stackrel{~}{\gamma }\left(v\right)=0$, where

$\stackrel{~}{\gamma }\left(v\right)=\frac{1+\mu \left(N-2\right)}{2}\left(N-2\right){\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1+\mu N}{2}{\int }_{{ℝ}^{N}}\left[NV\left(x\right)+\left(\nabla V\left(x\right),x\right)\right]{f}^{2}\left(v\right)𝑑x$$+\frac{\mu N}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x+\frac{\mu }{2}{\int }_{{ℝ}^{N}}\left(x\cdot {H}_{V}\left(x\right)\cdot x\right){f}^{2}\left(v\right)𝑑x-\left(1+\mu N\right)N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x.$

Recalling that $v\in \mathcal{𝒫}$, and substituting $\gamma \left(v\right)=0$ in the equation above, we get

$\stackrel{~}{\gamma }\left(v\right)=-\mu \left(N-2\right){\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{\mu N}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x+\frac{\mu }{2}{\int }_{{ℝ}^{N}}\left(x\cdot {H}_{V}\left(x\right)\cdot x\right){f}^{2}\left(v\right)𝑑x.$

Since v is a solution of equation (2.4), it satisfies $\stackrel{~}{\gamma }\left(v\right)=0$. This yields

$\mu \left(N-2\right){\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x=\frac{\mu N}{2}{\int }_{{ℝ}^{N}}\left[\left(\nabla V\left(x\right),x\right)+\frac{x\cdot {H}_{V}\left(x\right)\cdot x}{N}\right]{f}^{2}\left(v\right)𝑑x.$

From (V${}_{5}$) we get that, if $\mu <0$, the right-hand side of the above equation is nonnegative, while the left-hand side is negative. If $\mu >0$, one gets the same contradiction. Hence $\mu =0$. ∎

## 3 Proof of Theorem 1.1

In this section, we apply ideas similar to those employed in [16, 4, 17]. Set ${k}_{\mathrm{\infty }}\left(s\right):=g\left(f\left(s\right)\right){f}^{\prime }\left(s\right)-{V}_{\mathrm{\infty }}f\left(s\right){f}^{\prime }\left(s\right)$; then equation (1.4) becomes

$-\mathrm{\Delta }v={k}_{\mathrm{\infty }}\left(v\right).$(3.1)

Since equation (3.1) is a semilinear equation, we can use the conclusions in [16, 17]. Let

${K}_{\mathrm{\infty }}\left(s\right):={\int }_{0}^{s}{k}_{\mathrm{\infty }}\left(t\right)𝑑t=G\left(f\left(s\right)\right)-\frac{1}{2}{V}_{\mathrm{\infty }}{f}^{2}\left(s\right);$

then we can get the following lemmas. The proof of these lemmas can be found in [16]; we omit them.

#### Lemma 3.1.

Assume that $v\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ and ${\mathrm{\int }}_{{\mathrm{R}}^{N}}{K}_{\mathrm{\infty }}\mathit{}\mathrm{\left(}v\mathrm{\right)}\mathit{}𝑑x\mathrm{>}\mathrm{0}$. Then there exist unique ${t}_{\mathrm{1}}\mathrm{>}\mathrm{0}$ and ${t}_{\mathrm{2}}\mathrm{>}\mathrm{0}$ such that $v\mathrm{\left(}\mathrm{\cdot }\mathrm{/}{t}_{\mathrm{1}}\mathrm{\right)}\mathrm{\in }\mathcal{P}$ and $v\mathrm{\left(}\mathrm{\cdot }\mathrm{/}{t}_{\mathrm{2}}\mathrm{\right)}\mathrm{\in }{\mathcal{P}}_{\mathrm{\infty }}$.

If $v\mathrm{\in }\mathcal{P}$, then there exists ${t}_{v}\mathrm{>}\mathrm{0}$ such that $v\mathrm{\left(}\mathrm{\cdot }\mathrm{/}{t}_{v}\mathrm{\right)}\mathrm{\in }{\mathcal{P}}_{\mathrm{\infty }}$ and ${t}_{v}\mathrm{<}\mathrm{1}$.

If $w\mathrm{\in }{\mathcal{P}}_{\mathrm{\infty }}$, then there exists ${t}_{w}\mathrm{>}\mathrm{0}$ such that $w\mathrm{\left(}\mathrm{\cdot }\mathrm{/}{t}_{w}\mathrm{\right)}\mathrm{\in }\mathcal{P}$ and ${t}_{w}\mathrm{>}\mathrm{1}$.

#### Lemma 3.2.

Let $\mathrm{\Omega }\mathrm{=}\mathrm{\left\{}v\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}\mathrm{:}v\mathrm{\ne }\mathrm{0}\mathrm{,}{\mathrm{\int }}_{{\mathrm{R}}^{N}}{K}_{\mathrm{\infty }}\mathit{}\mathrm{\left(}v\mathrm{\right)}\mathit{}𝑑x\mathrm{>}\mathrm{0}\mathrm{\right\}}$. Then the function ${t}_{\mathrm{1}}\mathrm{:}\mathrm{\Omega }\mathrm{\to }{\mathrm{R}}^{\mathrm{+}}$ given by $v\mathrm{↦}{t}_{\mathrm{1}}\mathit{}\mathrm{\left(}v\mathrm{\right)}$ such that $v\mathrm{\left(}\mathrm{\cdot }\mathrm{/}{t}_{\mathrm{1}}\mathrm{\left(}v\mathrm{\right)}\mathrm{\right)}\mathrm{\in }\mathcal{P}$ is continuous.

#### Lemma 3.3.

Let $v\mathrm{\in }{\mathcal{P}}_{\mathrm{\infty }}$. Then for any $y\mathrm{\in }{\mathrm{R}}^{N}$, we have $v\mathrm{\left(}\mathrm{\cdot }\mathrm{-}y\mathrm{\right)}\mathrm{\in }{\mathcal{P}}_{\mathrm{\infty }}$. Moreover, there exists ${t}_{y}\mathrm{>}\mathrm{1}$ such that

$v\left(\frac{\cdot -y}{{t}_{y}}\right)\in \mathcal{𝒫}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\underset{|y|\to +\mathrm{\infty }}{lim}{t}_{y}=1.$

#### Lemma 3.4.

There holds ${\mathrm{sup}}_{y\mathrm{\in }{\mathrm{R}}^{N}}\mathit{}{t}_{y}\mathrm{:=}\overline{t}\mathrm{<}\mathrm{+}\mathrm{\infty }$ and $\overline{t}\mathrm{>}\mathrm{1}$.

#### Lemma 3.5.

There exists a real number $\stackrel{\mathrm{^}}{\sigma }\mathrm{>}\mathrm{0}$ such that ${\mathrm{inf}}_{v\mathrm{\in }\mathcal{P}}\mathit{}{\mathrm{\parallel }\mathrm{\nabla }\mathit{}v\mathrm{\parallel }}_{\mathrm{2}}\mathrm{\ge }\stackrel{\mathrm{^}}{\sigma }$.

#### Lemma 3.6.

There holds $p\mathrm{=}{\mathrm{inf}}_{v\mathrm{\in }\mathcal{P}}\mathit{}I\mathit{}\mathrm{\left(}v\mathrm{\right)}\mathrm{>}\mathrm{0}$ and $p\mathrm{=}{c}_{\mathrm{\infty }}$.

#### Proof of Theorem 1.1.

Arguing by contradiction, we suppose that there is $v\in {H}^{1}\left({ℝ}^{N}\right)$ such that $I\left(v\right)=p$ and ${I}^{\prime }\left(v\right)=0$. Then $v\in \mathcal{𝒫}$. By Lemma 3.1, there exists ${t}_{v}>0$ such that $v\left(\cdot /{t}_{v}\right)\in {\mathcal{𝒫}}_{\mathrm{\infty }}$ and ${t}_{v}<1$. From (2.3) and (V${}_{3}$) we deduce that

$p=I\left(v\right)=\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x-{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x$$=\frac{1}{N}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x-\frac{1}{2N}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x$$>\frac{{t}_{v}^{N-2}}{N}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x={I}_{\mathrm{\infty }}\left(v\left(\frac{x}{{t}_{v}}\right)\right)\ge {c}_{\mathrm{\infty }},$

which is a contradiction to Lemma 3.6. ∎

## 4 Proof of Theorem 1.2

This section is dedicated to proving the existence of a positive solution for equation (1.3). By the previous results, we should search for solutions which have energy levels above ${c}_{\mathrm{\infty }}$. Similarly to what was done in [16, 17], we start by showing that the min-max levels of the mountain pass theorem for the functionals I and ${I}_{\mathrm{\infty }}$ are equal.

#### Lemma 4.1.

There holds ${c}_{\mathrm{\infty }}\mathrm{=}c\mathrm{=}p$.

The proof is analogous to the proofs of [16, Lemmas 4.1 and 4.2]; we omit it.

#### Lemma 4.2.

For every $\zeta \mathrm{\in }\mathrm{\Gamma }$, there exists $s\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ such that $\zeta \mathit{}\mathrm{\left(}s\mathrm{\right)}$ intersects $\mathcal{P}$.

#### Proof.

By the proof of Lemma 2.2 (1), we learn that there exists $0<\rho <1$ such that $\gamma \left(v\right)>0$ if $0<\parallel v\parallel <\rho$. Furthermore, we observe that

$\gamma \left(v\right)=\frac{N-2}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{N}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left(v\right)𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x-N{\int }_{{ℝ}^{N}}G\left(f\left(v\right)\right)𝑑x$$=NI\left(v\right)-{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left(v\right)𝑑x.$

It follows from (V${}_{3}$) that

$\gamma \left(v\right)

Therefore, if $\zeta \in \mathrm{\Gamma }$, we have $\gamma \left(\zeta \left(0\right)\right)=0$ and $\gamma \left(\zeta \left(1\right)\right). Since $I\left(\zeta \left(1\right)\right)<0$, we conclude that there exists $s\in \left(0,1\right)$ such that $\gamma \left(\zeta \left(s\right)\right)=0$ for which $\parallel \zeta \left(s\right)\parallel >\rho$. The function $\zeta \left(s\right)$ satisfies $\zeta \left(s\right)\in \mathcal{𝒫}$, which shows that every path $\zeta \in \mathrm{\Gamma }$ intersects $\mathcal{𝒫}$. ∎

#### Lemma 4.3.

There exists a ${\mathrm{\left(}C\mathrm{\right)}}_{c}$ sequence $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}\mathrm{\subset }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ where

$c=\underset{\zeta \in \mathrm{\Gamma }}{inf}\underset{t\in \left[0,1\right]}{\mathrm{max}}I\left(\zeta \left(t\right)\right).$

The proof of Lemma 4.3 can be found in [16] (see also [17, 18]).

#### Lemma 4.4.

If $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}$ is a ${\mathrm{\left(}C\mathrm{\right)}}_{d}$ sequence with $d\mathrm{>}\mathrm{0}$, then it has a bounded subsequence.

#### Proof.

First of all, we observe that if a sequence $\left\{{v}_{n}\right\}\subset {H}^{1}\left({ℝ}^{N}\right)$ satisfies

${\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x+{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x\le {C}_{4}$

for some constant ${C}_{4}>0$, then the sequence $\left\{{v}_{n}\right\}$ is bounded in ${H}^{1}\left({ℝ}^{N}\right)$. For that, we simply need to demonstrate that ${\int }_{{ℝ}^{N}}{v}_{n}^{2}𝑑x$ is bounded. In fact, by Lemma 2.1 (9) and (V${}_{2}$), we observe that

${\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|\le 1\right\}}{v}_{n}^{2}𝑑x\le \frac{1}{{C}^{2}}{\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|\le 1\right\}}{f}^{2}\left({v}_{n}\right)𝑑x$$\le \frac{1}{{C}^{2}{V}_{\mathrm{\infty }}}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x\le \frac{{C}_{4}}{{C}^{2}{V}_{\mathrm{\infty }}}.$

Moreover, by the Sobolev inequality and Lemma 2.1 (9), one deduces

${\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|>1\right\}}{v}_{n}^{2}𝑑x\le {\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|>1\right\}}{v}_{n}^{{2}^{*}}𝑑x\le {C}_{5}{\left({\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|>1\right\}}{|\nabla {v}_{n}|}^{2}𝑑x\right)}^{\frac{{2}^{*}}{2}}$$\le {C}_{5}{\left({\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x\right)}^{\frac{{2}^{*}}{2}}\le {C}_{5}{C}_{4}^{\frac{{2}^{*}}{2}}.$

Hence there is a constant ${C}_{6}>0$ such that

${\int }_{{ℝ}^{N}}{v}_{n}^{2}𝑑x={\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|\le 1\right\}}{v}_{n}^{2}𝑑x+{\int }_{\left\{x\in {ℝ}^{N}:|{v}_{n}\left(x\right)|>1\right\}}{v}_{n}^{2}𝑑x\le {C}_{6}.$

Therefore, it remains to show that

${\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x+{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x$

is bounded.

Let $\left\{{v}_{n}\right\}\subset {H}^{1}\left({ℝ}^{N}\right)$ be an arbitrary Cerami sequence for I at level $d>0$, that is,

$I\left({v}_{n}\right)\to d \text{and} \left(1+\parallel {v}_{n}\parallel \right)\parallel {I}^{\prime }\left({v}_{n}\right)\parallel \to 0,$

namely

$\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x-{\int }_{{ℝ}^{N}}G\left(f\left({v}_{n}\right)\right)𝑑x=d+{o}_{n}\left(1\right),$(4.1)

and for any $\phi \in {H}^{1}\left({ℝ}^{N}\right)$,

$〈{I}^{\prime }\left({v}_{n}\right),\phi 〉={\int }_{{ℝ}^{N}}\nabla {v}_{n}\cdot \nabla \phi dx+{\int }_{{ℝ}^{N}}V\left(x\right)f\left({v}_{n}\right){f}^{\prime }\left({v}_{n}\right)\phi 𝑑x-{\int }_{{ℝ}^{N}}g\left(f\left({v}_{n}\right)\right){f}^{\prime }\left({v}_{n}\right)\phi 𝑑x={o}_{n}\left(1\right).$

Choosing

$\phi ={\phi }_{n}=\sqrt{1+2{f}^{2}\left({v}_{n}\right)}f\left({v}_{n}\right)=\frac{f\left({v}_{n}\right)}{{f}^{\prime }\left({v}_{n}\right)}$

from Lemma 2.1 (6), we get ${\parallel {\phi }_{n}\parallel }_{2}\le 2{\parallel {v}_{n}\parallel }_{2}$ and

$|\nabla {\phi }_{n}|=\left(1+\frac{2{f}^{2}\left({v}_{n}\right)}{1+2{f}^{2}\left({v}_{n}\right)}\right)|\nabla {v}_{n}|\le 2|\nabla {v}_{n}|.$

Thus there exists a constant ${C}_{7}>0$ such that $\parallel {\phi }_{n}\parallel \le {C}_{7}\parallel {v}_{n}\parallel$. Recalling that $\left\{{v}_{n}\right\}\subset {H}^{1}\left({ℝ}^{N}\right)$ is a Cerami sequence, we get

$〈{I}^{\prime }\left({v}_{n}\right),{\phi }_{n}〉={\int }_{{ℝ}^{N}}\left(1+\frac{2{f}^{2}\left({v}_{n}\right)}{1+2{f}^{2}\left({v}_{n}\right)}\right){|\nabla {v}_{n}|}^{2}𝑑x+{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x-{\int }_{{ℝ}^{N}}g\left(f\left({v}_{n}\right)\right)f\left({v}_{n}\right)𝑑x={o}_{n}\left(1\right).$(4.2)

Computing (4.1) $-$ $\frac{1}{4}$(4.2), we get

$d+{o}_{n}\left(1\right)=\frac{1}{4}{\int }_{{ℝ}^{N}}\frac{1}{1+2{f}^{2}\left({v}_{n}\right)}{|\nabla {v}_{n}|}^{2}𝑑x+\frac{1}{4}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x+{\int }_{{ℝ}^{N}}\left(\frac{1}{4}g\left(f\left({v}_{n}\right)\right)f\left({v}_{n}\right)-G\left(f\left({v}_{n}\right)\right)\right)𝑑x.$

Thanks to (2.2), we get

$\frac{1}{4}{\int }_{{ℝ}^{N}}\frac{1}{1+2{f}^{2}\left({v}_{n}\right)}{|\nabla {v}_{n}|}^{2}𝑑x+\frac{1}{4}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x\le d+{o}_{n}\left(1\right).$(4.3)

Denote ${w}_{n}=f\left({v}_{n}\right)$; then ${|\nabla {v}_{n}|}^{2}=\left(1+2{w}_{n}^{2}\right){|\nabla {w}_{n}|}^{2}$. We can rewrite (4.1) and (4.3) as follows:

$\frac{1}{2}{\int }_{{ℝ}^{N}}\left(1+2{w}_{n}^{2}\right){|\nabla {w}_{n}|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){w}_{n}^{2}𝑑x-{\int }_{{ℝ}^{N}}G\left({w}_{n}\right)𝑑x=d+{o}_{n}\left(1\right)$(4.4)

and

$\frac{1}{4}{\int }_{{ℝ}^{N}}{|\nabla {w}_{n}|}^{2}𝑑x+\frac{1}{4}{\int }_{{ℝ}^{N}}V\left(x\right){w}_{n}^{2}𝑑x\le d+{o}_{n}\left(1\right).$(4.5)

From (4.5) and (V${}_{2}$) we can see that $\left\{{w}_{n}\right\}$ is bounded in ${H}^{1}\left({ℝ}^{N}\right)$. It follows from (1.6) that

${\int }_{{ℝ}^{N}}G\left({w}_{n}\right)𝑑x\le {\int }_{{ℝ}^{N}}\left(\frac{\epsilon }{2}{|{w}_{n}|}^{2}+{C}_{\epsilon }{|{w}_{n}|}^{{2}^{*}}\right)𝑑x$$\le \frac{\epsilon }{2}{\int }_{{ℝ}^{N}}{|{w}_{n}|}^{2}𝑑x+{C}_{\epsilon }{S}^{-\frac{{2}^{*}}{2}}{\left({\int }_{{ℝ}^{N}}{|\nabla {w}_{n}|}^{2}\right)}^{\frac{{2}^{*}}{2}}$$\le {C}_{8}.$

By the above inequality and (4.4), one has

$\frac{1}{2}{\int }_{{ℝ}^{N}}\left(1+2{w}_{n}^{2}\right){|\nabla {w}_{n}|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){w}_{n}^{2}𝑑x\le {C}_{8}+d+{o}_{n}\left(1\right),$

namely

$\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x\le {C}_{8}+d+{o}_{n}\left(1\right).\mathit{∎}$

#### Lemma 4.5 (Splitting (see [33])).

Let $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}\mathrm{\subset }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ be a bounded sequence such that

$I\left({v}_{n}\right)\to c\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\left(1+\parallel {v}_{n}\parallel \right)\parallel {I}^{\prime }\left({v}_{n}\right)\parallel \to 0.$

Then there exists (if necessary, replace $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}$ by a subsequence) a solution $\overline{v}$ of equation (1.3), a number $k\mathrm{\in }\mathrm{N}\mathrm{\cup }\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$, k functions ${v}^{\mathrm{1}}\mathrm{,}{v}^{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{v}^{k}$ and k sequences of points $\mathrm{\left\{}{y}_{n}^{j}\mathrm{\right\}}\mathrm{\subset }{\mathrm{R}}^{N}$, $\mathrm{1}\mathrm{\le }j\mathrm{\le }k$, satisfying the following properties:

• (1)

${v}_{n}\to \overline{v}$ in ${H}^{1}\left({ℝ}^{N}\right)$ or

• (2)

${v}^{j}$ are nontrivial solutions of equation ( 1.4 );

• (3)

$|{y}_{n}^{j}|\to \mathrm{\infty }$ and $|{y}_{n}^{j}-{y}_{n}^{i}|\to \mathrm{\infty }$, $i\ne j$ ;

• (4)

${v}_{n}-{\sum }_{i=1}^{k}{v}^{i}\left(x-{y}_{n}^{i}\right)\to \overline{v}$ ;

• (5)

$I\left({v}_{n}\right)\to I\left(\overline{v}\right)+{\sum }_{i=1}^{k}{I}_{\mathrm{\infty }}\left({v}^{i}\right)$.

#### Corollary 4.6.

If $I\mathit{}\mathrm{\left(}{v}_{n}\mathrm{\right)}\mathrm{\to }{c}_{\mathrm{\infty }}$ and $\mathrm{\left(}\mathrm{1}\mathrm{+}\mathrm{\parallel }{v}_{n}\mathrm{\parallel }\mathrm{\right)}\mathit{}\mathrm{\parallel }{I}^{\mathrm{\prime }}\mathit{}\mathrm{\left(}{v}_{n}\mathrm{\right)}\mathrm{\parallel }\mathrm{\to }\mathrm{0}$, then either $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}$ is relatively compact or the splitting lemma holds with $k\mathrm{=}\mathrm{1}$ and $\overline{v}\mathrm{=}\mathrm{0}$.

Let us set

Then we have the following lemma.

#### Lemma 4.7.

Assume that ${c}_{\mathrm{\infty }}$ is an isolated radial critical level for ${I}_{\mathrm{\infty }}$. Then ${c}_{\mathrm{♯}}\mathrm{>}{c}_{\mathrm{\infty }}$ and I satisfies condition (3) at level $d\mathrm{\in }\mathrm{\left(}{c}_{\mathrm{\infty }}\mathrm{,}\mathrm{min}\mathit{}\mathrm{\left\{}{c}_{\mathrm{♯}}\mathrm{,}\mathrm{2}\mathit{}{c}_{\mathrm{\infty }}\mathrm{\right\}}\mathrm{\right)}$. Assume now that the limiting equation (1.4) admits a unique positive radial solution. Then I satisfies condition (3) at level $d\mathrm{\in }\mathrm{\left(}{c}_{\mathrm{\infty }}\mathrm{,}\mathrm{2}\mathit{}{c}_{\mathrm{\infty }}\mathrm{\right)}$.

The proof is analogous to [18, Lemma 5.9]; we omit it.

#### Lemma 4.8.

If $I\mathit{}\mathrm{\left(}{v}_{n}\mathrm{\right)}\mathrm{\to }d\mathrm{>}\mathrm{0}$ and $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}\mathrm{\subset }\mathcal{P}$, then the sequence $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}$ is bounded.

#### Proof.

Since $I\left({v}_{n}\right)\to d>0$ and $\left\{{v}_{n}\right\}\subset \mathcal{𝒫}$, we get

$d+1>I\left({v}_{n}\right)=\frac{1}{N}{\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x-\frac{1}{2N}{\int }_{{ℝ}^{N}}\left(\nabla V\left(x\right),x\right){f}^{2}\left({v}_{n}\right)𝑑x\ge \frac{1}{N}{\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x,$

where we also used (V${}_{3}$). Therefore ${\parallel \nabla {v}_{n}\parallel }_{2}$ is bounded. By the Sobolev inequality, the sequence ${\parallel {v}_{n}\parallel }_{{2}^{*}}$ is also bounded.

It follows from (V${}_{2}$), (1.6), (2.1), and Lemma 2.1 (3) and (7) that

$d+1\ge I\left({v}_{n}\right)=\frac{1}{2}{\int }_{{ℝ}^{N}}{|\nabla {v}_{n}|}^{2}𝑑x+\frac{1}{2}{\int }_{{ℝ}^{N}}V\left(x\right){f}^{2}\left({v}_{n}\right)𝑑x-{\int }_{{ℝ}^{N}}G\left(f\left({v}_{n}\right)\right)𝑑x$$\ge \frac{1}{2}{\parallel \nabla {v}_{n}\parallel }_{2}^{2}+\frac{{V}_{\mathrm{\infty }}}{2}{\int }_{{ℝ}^{N}}{f}^{2}\left({v}_{n}\right)𝑑x-{\int }_{{ℝ}^{N}}\left(\frac{\epsilon }{2}{|f\left({v}_{n}\right)|}^{2}+{C}_{\epsilon }{|f\left({v}_{n}\right)|}^{2\cdot {2}^{*}}\right)𝑑x.$$\ge \frac{1}{2}{\parallel \nabla {v}_{n}\parallel }_{2}^{2}+\frac{{V}_{\mathrm{\infty }}}{2}{\int }_{{ℝ}^{N}}{C}_{1}\left({v}_{n}^{2}-{|{v}_{n}|}^{{2}^{*}}\right)𝑑x-{\int }_{{ℝ}^{N}}\left(\frac{\epsilon }{2}{v}_{n}^{2}+{2}^{\frac{{2}^{*}}{2}}{C}_{\epsilon }{|{v}_{n}|}^{{2}^{*}}\right)𝑑x.$$=\frac{1}{2}{\parallel \nabla {v}_{n}\parallel }_{2}^{2}+\left(\frac{{V}_{\mathrm{\infty }}{C}_{1}}{2}-\frac{\epsilon }{2}\right){\int }_{{ℝ}^{N}}{v}_{n}^{2}𝑑x-\left(\frac{{V}_{\mathrm{\infty }}{C}_{1}}{2}+{2}^{\frac{{2}^{*}}{2}}{C}_{\epsilon }\right){\int }_{{ℝ}^{N}}{|{v}_{n}|}^{{2}^{*}}𝑑x.$

Since ${\parallel \nabla {v}_{n}\parallel }_{2}$ and ${\parallel {v}_{n}\parallel }_{{2}^{*}}$ are bounded, ${\parallel {v}_{n}\parallel }_{2}$ is bounded as well. Hence $\left\{{v}_{n}\right\}$ is bounded in ${H}^{1}\left({ℝ}^{N}\right)$. ∎

#### Definition 4.9.

Define the barycenter function of a given function $u\in {H}^{1}\left({ℝ}^{N}\right)\setminus \left\{0\right\}$ by setting

$\mu \left(u\right)\left(x\right)=\frac{1}{|{B}_{1}|}{\int }_{{B}_{1}\left(x\right)}|u\left(y\right)|𝑑y,$

with $\mu \left(u\right)\in {L}^{\mathrm{\infty }}\left({ℝ}^{N}\right)$ and μ is a continuous function. Subsequently, take

$\stackrel{^}{u}\left(x\right)={\left[\mu \left(u\right)\left(x\right)-\frac{1}{2}\mathrm{max}\mu \left(u\right)\right]}^{+}.$

It follows that $\stackrel{^}{u}\in {C}_{0}\left({ℝ}^{N}\right)$. Now we define the barycenter of u by

$\beta \left(u\right)=\frac{1}{{\parallel \stackrel{^}{u}\parallel }_{1}}{\int }_{{ℝ}^{N}}x\stackrel{^}{u}\left(x\right)𝑑x\in {ℝ}^{N}.$

Then $\beta \left(u\right)$ is well defined since $\stackrel{^}{u}$ has compact support.

The function $\beta \left(u\right)$ satisfies the following properties:

• (1)

β is a continuous function in ${H}^{1}\left({ℝ}^{N}\right)\setminus \left\{0\right\}$;

• (2)

if u is radial, then $\beta \left(u\right)=0$;

• (3)

if $y\in {ℝ}^{N}$ is given and if we define ${u}_{y}\left(x\right):=u\left(x-y\right)$, then $\beta \left({u}_{y}\right)=\beta \left(u\right)+y$.

We shall also need the following lemma.

#### Lemma 4.10.

Assume that $\mathrm{\left\{}{u}_{n}\mathrm{\right\}}\mathrm{,}\mathrm{\left\{}{v}_{n}\mathrm{\right\}}\mathrm{\subset }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{N}\mathrm{\right)}$ are such that $\mathrm{\parallel }{u}_{n}\mathrm{-}{v}_{n}\mathrm{\parallel }\mathrm{\to }\mathrm{0}$ and ${I}^{\mathrm{\prime }}\mathit{}\mathrm{\left(}{v}_{n}\mathrm{\right)}\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$, where $\mathrm{\left\{}{v}_{n}\mathrm{\right\}}$ is bounded. Then ${I}^{\mathrm{\prime }}\mathit{}\mathrm{\left(}{u}_{n}\mathrm{\right)}\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$.

#### Proof.

We simply observe that

${f}^{\prime }\left(t\right)=\frac{1}{{\left(1+2{f}^{2}\left(t\right)\right)}^{1/2}},$

thus by Lemma 2.1 (10) we have

$|{f}^{\prime \prime }\left(t\right)|=|\frac{-2f\left(t\right){f}^{\prime }\left(t\right)}{{\left(1+2{f}^{2}\left(t\right)\right)}^{3/2}}|\le |2f\left(t\right){f}^{\prime }\left(t\right)|\le \sqrt{2}.$(4.6)

Since $g\in \mathrm{Lip}\left({ℝ}^{+},{ℝ}^{+}\right)$, we find that there is a constant ${C}_{9}>0$ such that

Therefore,

$|g\left(f\left({u}_{n}\right)\right){f}^{\prime }\left({u}_{n}\right)-g\left(f\left({v}_{n}\right)\right){f}^{\prime }\left({v}_{n}\right)|=|\left[g\left(f\left({u}_{n}\right)\right)-g\left(f\left({v}_{n}\right)\right)\right]{f}^{\prime }\left({u}_{n}\right)-g\left(f\left({v}_{n}\right)\right)\left[{f}^{\prime }\left({v}_{n}\right)-{f}^{\prime }\left({u}_{n}\right)\right]|$$\le {C}_{9}|f\left({u}_{n}\right)-f\left({v}_{n}\right)||{f}^{\prime }\left({u}_{n}\right)|+{C}_{9}|f\left({v}_{n}\right)-0||{f}^{\prime }\left({v}_{n}\right)-{f}^{\prime }\left({u}_{n}\right)|$$\le {C}_{9}|{f}^{\prime }\left({\xi }_{n}\right)||{u}_{n}-{v}_{n}||{f}^{\prime }\left({u}_{n}\right)|+{C}_{9}|{v}_{n}||{f}^{\prime \prime }\left({\xi }_{n}^{\prime }\right)||{v}_{n}-{u}_{n}|$$\le {C}_{9}|{u}_{n}-{v}_{n}|+\sqrt{2}{C}_{9}|{v}_{n}||{v}_{n}-{u}_{n}|,$

where we also used the mean value theorem, Lemma 2.1 (2) and (3) and (4.6). Hence, by using the Hölder inequality, one has

${\int }_{{ℝ}^{N}}|\left(g\left(f\left({u}_{n}\right)\right){f}^{\prime }\left({u}_{n}\right)-g\left(f\left({v}_{n}\right)\right){f}^{\prime }\left({v}_{n}\right)\right)\phi |𝑑x\le {C}_{9}{\int }_{{ℝ}^{N}}|{u}_{n}-{v}_{n}||\phi |𝑑x+\sqrt{2}{C}_{9}{\int }_{{ℝ}^{N}}|{v}_{n}||{v}_{n}-{u}_{n}||\phi |𝑑x$$\le {C}_{9}{\parallel {u}_{n}-{v}_{n}\parallel }_{2}{\parallel \phi \parallel }_{2}+\sqrt{2}{C}_{9}{\parallel {v}_{n}\phi \parallel }_{2}{\parallel {v}_{n}-{u}_{n}\parallel }_{2}$(4.7)

Since the function $f\left(s\right){f}^{\prime }\left(s\right)$ is continuous and $V\left(x\right)$ satisfies (V${}_{2}$), we can conclude that

(4.8)

We also find that

(4.9)

It follows from (4.7), (4.8) and (4.9) that

$〈{I}^{\prime }\left({u}_{n}\right)-{I}^{\prime }\left({v}_{n}\right),\phi 〉={\int }_{{ℝ}^{N}}\nabla \left({u}_{n}-{v}_{n}\right)\nabla \phi dx+{\int }_{{ℝ}^{N}}V\left(x\right)\left(\left(f\left({u}_{n}\right){f}^{\prime }\left({u}_{n}\right)-f\left({v}_{n}\right){f}^{\prime }\left({v}_{n}\right)\right)\phi dx$$-{\int }_{{ℝ}^{N}}\left(g\left(f\left({u}_{n}\right)\right){f}^{\prime }\left({u}_{n}\right)-g\left(f\left({v}_{n}\right)\right){f}^{\prime }\left({v}_{n}\right)\right)\phi 𝑑x$

Hence ${I}^{\prime }\left({u}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$. ∎

We define

It is clear that $b\ge {c}_{\mathrm{\infty }}$; moreover, we have the following lemma.

#### Lemma 4.11.

There holds $b\mathrm{>}{c}_{\mathrm{\infty }}$.

#### Proof.

Suppose $b={c}_{\mathrm{\infty }}$. By the definition of b, there is a minimizing sequence

such that $I\left({v}_{n}\right)\to b>0$. By Lemma 4.8, $\left\{{v}_{n}\right\}$ is bounded. Since $b={c}_{\mathrm{\infty }}=p$ by Lemma 4.1, $\left\{{v}_{n}\right\}$ is also a minimizing sequence of I on $\mathcal{𝒫}$. By Ekeland’s variational principle [33, Theorem 8.5], there is another sequence $\left\{{\stackrel{~}{v}}_{n}\right\}\subset \mathcal{𝒫}$ such that $I\left({\stackrel{~}{v}}_{n}\right)\to p$, ${{I}^{\prime }|}_{\mathcal{𝒫}}\left({\stackrel{~}{v}}_{n}\right)\to 0$ and $\parallel {\stackrel{~}{v}}_{n}-{v}_{n}\parallel \to 0$ as $n\to \mathrm{\infty }$. Now we prove that ${I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$. Indeed, if ${I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)$ does not go to zero, that means there exist ${\epsilon }_{0}>0$ and a subsequence denoted also by ${\stackrel{~}{v}}_{n}$ such that $\parallel {I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)\parallel >{\epsilon }_{0}$. Arguing as in the proof of Lemma 4.10, we can deduce that there is a positive constant ${C}_{10}$ such that

Thus, if

$\parallel {\stackrel{~}{v}}_{n}-v\parallel <\frac{\stackrel{~}{\delta }}{{C}_{10}}:=3\delta ,$

then

$\parallel {I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)-{I}^{\prime }\left(v\right)\parallel <\stackrel{~}{\delta }.$

This yields

${\epsilon }_{0}-\stackrel{~}{\delta }<\parallel {I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)\parallel -\stackrel{~}{\delta }<\parallel {I}^{\prime }\left(v\right)\parallel .$

For $\stackrel{~}{\delta }>0$ sufficiently small, we have $\lambda :={\epsilon }_{0}-\stackrel{~}{\delta }>0$, and for all $v\in {B}_{3\delta }\left({\stackrel{~}{v}}_{n}\right)$, one has $\parallel {I}^{\prime }\left(v\right)\parallel >\lambda$. Now let $\epsilon :=\mathrm{min}\left\{p/2,\left(\lambda \delta \right)/8\right\}$ and $S:=\left\{{\stackrel{~}{v}}_{n}\right\}$. By [33, Lemma 2.3], there is a deformation η on the level p, taking all the points of ${S}_{\delta }$ to the level $p-\epsilon$.

Moreover, for n sufficiently large,

$\underset{t>0}{\mathrm{max}}I\left(\eta \left(1,{\stackrel{~}{v}}_{n}\left(\frac{\cdot }{t}\right)\right)\right)\le p-\epsilon ,$

because $\left\{{\stackrel{~}{v}}_{n}\right\}$ is a minimizing sequence, $I\left({\stackrel{~}{v}}_{n}\right)\le p+\epsilon /2$, for n sufficiently large, and since $\left\{{\stackrel{~}{v}}_{n}\right\}\subset \mathcal{𝒫}$, we have

$\underset{t>0}{\mathrm{max}}I\left({\stackrel{~}{v}}_{n}\left(\frac{\cdot }{t}\right)\right)=I\left({\stackrel{~}{v}}_{n}\right)\to p.$

On the other hand, ${\zeta }_{0}\left(t\right):=\eta \left(1,{\stackrel{~}{v}}_{n}\left(\cdot /Mt\right)\right)$ is a path in Γ for M and n large enough, hence

$c\le \underset{t\in \left[0,1\right]}{\mathrm{max}}I\left(\eta \left(1,{\stackrel{~}{v}}_{n}\left(\frac{\cdot }{Mt}\right)\right)\right)=\underset{t>0}{\mathrm{max}}I\left(\eta \left(1,{\stackrel{~}{v}}_{n}\left(\frac{\cdot }{t}\right)\right)\right)\le p-\epsilon

which is a contradiction to $p=c$, provided by Lemma 4.1; hence ${I}^{\prime }\left({\stackrel{~}{v}}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$. By Lemma 4.10, we get ${I}^{\prime }\left({v}_{n}\right)\to 0$ as $n\to \mathrm{\infty }$. Hereafter, the sequence $\left\{{v}_{n}\right\}$ satisfies the assumptions of Corollary 4.6. Since $p={c}_{\mathrm{\infty }}$ and p is not attained by Theorem 1.1, the splitting lemma holds with $k=1$. This yields

${v}_{n}\left(x\right)={v}^{1}\left(x-{y}_{n}\right)+{o}_{n}\left(1\right),$

where ${y}_{n}\in {ℝ}^{N}$, $|{y}_{n}|\to \mathrm{\infty }$ and ${v}^{1}$ is a solution of equation (1.4). Making a translation, we obtain

${v}_{n}\left(x+{y}_{n}\right)={v}^{1}\left(x\right)+{o}_{n}\left(1\right).$

Calculating the barycenter function on both sides, we have

$\beta \left({v}_{n}\left(x+{y}_{n}\right)\right)=\beta \left({v}_{n}\right)-{y}_{n},\beta \left({v}^{1}\left(x\right)+{o}_{n}\left(1\right)\right)=\beta \left({v}^{1}\left(x\right)\right)+{o}_{n}\left(1\right),$

where the first equality comes from property (3) of the barycenter function β and the second one due to the continuity of β. Since $\beta \left({v}_{n}\right)=0$, $|{y}_{n}|\to \mathrm{\infty }$ and $\beta \left({v}^{1}\left(x\right)\right)=0$, we arrive at a contradiction, yielding $b>{c}_{\mathrm{\infty }}$. ∎

Let us consider the positive, radially symmetric, ground state solution $w\in {H}^{1}\left({ℝ}^{N}\right)$ of equation (1.4). We define the operator $\mathrm{\Pi }:{ℝ}^{N}\to \mathcal{𝒫}$ by

$\mathrm{\Pi }\left[y\right]\left(x\right)=w\left(\frac{x-y}{{t}_{y}}\right),$

where ${t}_{y}$ is the real number t which projects $w\left(\cdot -y\right)$ onto the Pohozaev manifold $\mathcal{𝒫}$. Since ${t}_{y}$ is unique and ${t}_{y}\left(w\left(\cdot -y\right)\right)$ is a continuous function of $w\left(\cdot -y\right)$, we have that Π is a continuous function of y.

The following lemma describes some properties of the operator Π; its proof can be found in [16, 17].

#### Lemma 4.12.

$\beta \left(\mathrm{\Pi }\left[y\right]\left(x\right)\right)=y$ and $I\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathit{}\mathrm{\left[}y\mathrm{\right]}\mathrm{\right)}\mathrm{\to }{c}_{\mathrm{\infty }}$ as $\mathrm{|}y\mathrm{|}\mathrm{\to }\mathrm{\infty }$.

#### Lemma 4.13.

Assume that

• (V6)

there holds

${\parallel {V}_{\mathrm{\infty }}-V\parallel }_{\mathrm{\infty }}<2\frac{\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}-{c}_{\mathrm{\infty }}}{{\overline{t}}^{N}{\parallel w\parallel }_{2}^{2}},$

where $\overline{t}={sup}_{y\in {ℝ}^{N}}{t}_{y}$.

Then $I\mathit{}\mathrm{\left(}\mathrm{\Pi }\mathit{}\mathrm{\left[}y\mathrm{\right]}\mathrm{\right)}\mathrm{<}\mathrm{min}\mathit{}\mathrm{\left\{}{c}_{\mathrm{♯}}\mathrm{,}\mathrm{2}\mathit{}{c}_{\mathrm{\infty }}\mathrm{\right\}}$.

#### Proof.

Since ${I}_{\mathrm{\infty }}$ is translation invariant, the maximum of $t\to {I}_{\mathrm{\infty }}\left(w\left(\cdot /t\right)\right)$ is attained at $t=1$ and ${t}_{y}>1$. It follows from (V${}_{6}$) and Lemma 2.1 (3) that

$I\left(\mathrm{\Pi }\left[y\right]\right)={I}_{\mathrm{\infty }}\left(\mathrm{\Pi }\left[y\right]\right)+I\left(\mathrm{\Pi }\left[y\right]\right)-{I}_{\mathrm{\infty }}\left(\mathrm{\Pi }\left[y\right]\right)$$\le {I}_{\mathrm{\infty }}\left(w\right)+\frac{1}{2}{\int }_{{ℝ}^{N}}\left(V\left(x\right)-{V}_{\mathrm{\infty }}\right){f}^{2}\left(\mathrm{\Pi }\left[y\right]\right)𝑑x$$<{c}_{\mathrm{\infty }}+\frac{\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}-{c}_{\mathrm{\infty }}}{{\overline{t}}^{N}{\parallel w\parallel }_{2}^{2}}{\int }_{{ℝ}^{N}}{\left(\mathrm{\Pi }\left[y\right]\right)}^{2}𝑑x$$={c}_{\mathrm{\infty }}+\frac{\left(\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}-{c}_{\mathrm{\infty }}\right){t}_{y}^{N}}{{\overline{t}}^{N}{\parallel w\parallel }_{2}^{2}}{\parallel w\parallel }_{2}^{2}$$\le \mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\},$

which concludes the proof. ∎

#### Remark 4.14.

Replacing (V${}_{6}$) with

${\parallel {V}_{\mathrm{\infty }}-V\parallel }_{\mathrm{\infty }}<\frac{2{c}_{\mathrm{\infty }}}{{\overline{t}}^{N}{\parallel w\parallel }_{2}^{2}}$

yields $I\left(\mathrm{\Pi }\left[y\right]\right)<2{c}_{\mathrm{\infty }}$.

#### Definition 4.15.

Let S be a closed subset of a Banach space X and let Q be a submanifold of X with relative boundary $\partial Q$. We say that S and $\partial Q$ link if the following facts hold:

• (1)

$S\cap \partial Q=\mathrm{\varnothing }$;

• (2)

for any $h\in {C}^{0}\left(X,X\right)$ such that ${h\mid }_{\partial Q}=\mathrm{id}$ there holds $h\left(Q\right)\cap S\ne \mathrm{\varnothing }$.

Moreover, if S and Q are as stated above and B is a subset of ${C}^{0}\left(X,X\right)$, then S and $\partial Q$ link with respect to B if (1) and (2) hold for any $h\in B$.

Suppose that $I\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}X\mathrm{,}\mathrm{R}\mathrm{\right)}$ is a functional satisfying the (C) condition. Consider a closed subset $S\mathrm{\subset }X$ and a submanifold $Q\mathrm{\subset }X$ with relative boundary $\mathrm{\partial }\mathit{}Q$. In addition, suppose the following:

• (1)

S and $\partial Q$ link;

• (2)

$\alpha ={inf}_{u\in S}I\left(u\right)>{sup}_{u\in \partial Q}I\left(u\right)={\alpha }_{0}$ ;

• (3)

${sup}_{u\in Q}I\left(u\right)<+\mathrm{\infty }$.

If $B\mathrm{=}\mathrm{\left\{}h\mathrm{\in }{C}^{\mathrm{0}}\mathit{}\mathrm{\left(}X\mathrm{,}X\mathrm{\right)}\mathrm{:}{h\mathrm{\mid }}_{\mathrm{\partial }\mathit{}Q}\mathrm{=}\mathrm{id}\mathrm{\right\}}$, then the real number $\tau \mathrm{=}{\mathrm{inf}}_{h\mathrm{\in }B}\mathit{}{\mathrm{sup}}_{u\mathrm{\in }Q}\mathit{}I\mathit{}\mathrm{\left(}h\mathit{}\mathrm{\left(}u\mathrm{\right)}\mathrm{\right)}$ defines a critical value of I with $\tau \mathrm{\ge }\alpha$.

Now we are ready to prove our main existence result.

#### Proof of Theorem 1.2.

It follows from (V${}_{2}$) that ${I}_{\mathrm{\infty }}\left(v\right) for all $v\in {H}^{1}\left({ℝ}^{N}\right)\setminus \left\{0\right\}$, hence ${I}_{\mathrm{\infty }}\left(\mathrm{\Pi }\left[y\right]\right) for any $y\in {ℝ}^{N}$. From Lemma 4.11 and Lemma 4.12 we have $b>{c}_{\mathrm{\infty }}$ and $I\left(\mathrm{\Pi }\left[y\right]\right)\to {c}_{\mathrm{\infty }}$ as $|y|\to \mathrm{\infty }$, hence there is $\overline{\rho }>0$ such that for all $\rho \ge \overline{\rho }$,

${c}_{\mathrm{\infty }}<\underset{|y|=\rho }{\mathrm{max}}I\left(\mathrm{\Pi }\left[y\right]\right)(4.10)

In order to apply the linking theorem, we take

We will show that the conditions of Theorem 4.16 are satisfied.

(1) Since $\beta \left(\mathrm{\Pi }\left[y\right]\left(x\right)\right)=y$, from Lemma 4.12 we have that $S\cap \partial Q=\mathrm{\varnothing }$, because if $v\in S$, then $\beta \left(v\right)=0$, and if $v\in \partial Q$, then $\beta \left(v\right)=y\ne 0$ due to the equality $|y|=\overline{\rho }$. Now we show that $h\left(Q\right)\cap S\ne \mathrm{\varnothing }$, for any $h\in \mathcal{ℋ}$, where

$\mathcal{ℋ}=\left\{h\in C\left(Q,\mathcal{𝒫}\right):{h\mid }_{\partial Q}=\mathrm{id}\right\}$

Given $h\in \mathcal{ℋ}$, let us define $T:\overline{{B}_{\overline{\rho }}}\to {ℝ}^{N}$ for $T\left(y\right)=\beta \circ h\circ \mathrm{\Pi }\left[y\right]$. The function T is continuous, because it is the composition of continuous functions. Moreover, for any $|y|=\overline{\rho }$, we have that $\mathrm{\Pi }\left[y\right]\in \partial Q$, thus $h\circ \mathrm{\Pi }\left[y\right]=\mathrm{\Pi }\left[y\right]$, because ${h\mid }_{\partial Q}=\mathrm{id}$. Hence from Lemma 4.12 we have $T\left(y\right)=y$. By the fixed point theorem of Brouwer, we conclude that there exists $\stackrel{~}{y}\in {B}_{\overline{\rho }}$ such that $T\left(\stackrel{~}{y}\right)=0$, which implies $h\left(\mathrm{\Pi }\left[\stackrel{~}{y}\right]\right)\in S$. Therefore $h\left(Q\right)\cap S\ne \mathrm{\varnothing }$. From Definition 4.15, we know that S and $\partial Q$ link, i.e. relation (1) is proved.

(2) From the definitions of b and Q and inequality (4.10) we may write

$b=\underset{S}{inf}I>\underset{\partial Q}{\mathrm{max}}I,$

which implies relation (2).

(3) Let us define

$d=\underset{h\in \mathcal{ℋ}}{inf}\underset{v\in Q}{\mathrm{max}}I\left(h\left(v\right)\right).$

Then we have $d\ge b$. Indeed, we have already proved that $h\left(Q\right)\cap S\ne \mathrm{\varnothing }$ for all $h\in \mathcal{ℋ}$. If h is fixed, then there exists $w\in S$ such that w belongs to $h\left(Q\right)$, which means that $w=h\left(u\right)$ for some $u\in \mathrm{\Pi }\left(\overline{{B}_{\overline{\rho }}}\right)=Q$. Therefore,

$I\left(w\right)\ge \underset{v\in S}{inf}I\left(v\right),\underset{v\in Q}{\mathrm{max}}I\left(h\left(v\right)\right)\ge I\left(h\left(u\right)\right).$

This gives

$\underset{v\in Q}{\mathrm{max}}I\left(h\left(v\right)\right)\ge I\left(h\left(u\right)\right)=I\left(w\right)\ge \underset{v\in S}{inf}I\left(v\right)=b,$

and hence

$d=\underset{h\in \mathcal{ℋ}}{inf}\underset{v\in Q}{\mathrm{max}}I\left(h\left(v\right)\right)\ge b.$

In particular, it follows that $d>{c}_{\mathrm{\infty }}$ because from Lemma 4.11 we know that $b>{c}_{\mathrm{\infty }}$. Furthermore, if we take $h=\mathrm{id}$, then using Lemma 4.13, we get

$\underset{h\in \mathcal{ℋ}}{inf}\underset{v\in Q}{\mathrm{max}}I\left(h\left(v\right)\right)<\underset{v\in Q}{\mathrm{max}}I\left(v\right)<\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}.$(4.11)

Hence we can deduce that $d<\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}$; furthermore, we have $d\in \left({c}_{\mathrm{\infty }},\mathrm{min}\left\{{c}_{\mathrm{♯}},2{c}_{\mathrm{\infty }}\right\}\right)$. Thanks to Lemma 4.7, we get that the (C) condition is satisfied at level d. Also, inequality (4.11) tells us that relation (3) is satisfied.

From (1), (2) and (3) above we can apply the linking theorem and conclude that d is a critical level for the functional I. Hence there exists a nontrivial solution $v\in {H}^{1}\left({ℝ}^{N}\right)$ of equation (1.3). Reasoning as usual, because of the hypotheses on g and f, and using the maximum principle, we conclude that v is positive and Theorem 1.2 is proved. ∎

## Acknowledgements

The authors would like to thank the referee for his/her useful suggestions.

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Revised: 2017-02-13

Accepted: 2017-02-20

Published Online: 2017-05-11

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11471267

This paper was supported by the National Natural Science Foundation of China (No. 11471267).

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 323–338, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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