Infinitely many solutions for a class of fractional Orlicz-Sobolev Schr\"odinger equations

TL;DR

This paper proves the existence of infinitely many solutions for a class of fractional Orlicz-Sobolev Schrödinger equations using variational methods and a new compact embedding theorem.

Contribution

It introduces a new compact embedding theorem for fractional Orlicz-Sobolev spaces and applies it to establish multiple solutions for the Schrödinger equations.

Findings

Established a new compact embedding theorem.

Proved existence of infinitely many solutions.

Applied variational methods with the Fountain theorem.

Abstract

In the present paper, we deal with a new compact embedding theorem for a subspace of the new fractional Orlicz-Sobolev spaces. We also establish some useful inequalities which yields to apply the variational methods. Using these abstract results, we study the existence of infinitely many nontrivial solutions for a class of fractional Orlicz-Sobolev Schr\"odinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}+V(x)m(u)u=f(x,u),\ x\in\mathbb{R}^{N},$$ where $s\in ]0,1[$, $N\geq2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u| \rightarrow\infty$. The proof is based on the variant Fountain theorem established by Zou.

Equations

$$(-△{)}_m^{s}u+V(x)m(u)u=f(x,u),x\in {\mathbb{R}}^N,$$

$$(-△{)}_m^{s}u+V(x)m(u)u=f(x,u),x\in {\mathbb{R}}^N,$$

$$-△u+V(x)u=f(x,u),x\in {\mathbb{R}}^N.$$

$$\underset{|y|\to +\infty }{\lim }meas(\{x\in {\mathbb{R}}^N:|x-y|\le \nu ,V(x)\le L\})=0,\forall L>0,$$

$$-{△}_{m}u+V(x)m(u)u=f(x,u),x\in {\mathbb{R}}^N,$$

$$(-△{)}_p^{s}u+V(x)|u{|}^{p-2}u=f(x,u),x\in {\mathbb{R}}^N,$$

$$W^{s,M}(\Omega)=\bigg{\{}u\in L^{M}(\Omega):\ \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}M\bigg{(}\frac{u(x)-u(y)}{|x-y|^{s}}\bigg{)}\frac{dxdy}{|x-y|^{N}}<\infty\bigg{\}}.$$

$$W^{s,M}(\Omega)=\bigg{\{}u\in L^{M}(\Omega):\ \exists\lambda>0/\ \int_{\Omega}\int_{\Omega}M\bigg{(}\frac{\lambda(u(x)-u(y))}{|x-y|^{s}M^{-1}(|x-y|^{N})}\bigg{)}dxdy<\infty\bigg{\}}.$$

$$l\le r<l^{\ast }=\frac{Nl}{N-l}\text{and}l\le \frac{m(t)t^2}{M(t)}\le r,\forall t\ne 0,$$

$$M(t)={\int }_0^{|t|}m(s)sds.$$

$$M_{\ast }(t)={\int }_0^{|t|}{m}_{\ast }(s)sds.$$

$$C|t{|}^{\mu }\le M_{\ast }(at),\forall a,t\ge 0,\forall 1<\mu \le r,$$

$$M(t)={\int }_0^{|t|}m(s)ds,$$

$$\overline{M}(t)={\int }_0^{|t|}\overline{m}(s)ds,$$

$$st\le M(s)+\overline{M}(t),$$

$$M(2t)\le KM(t),\forall t\ge 0,$$

$$M(st)\le K_{s}M(t),\forall t\ge 0.$$

$$B(x)\le A(ax),x\ge x_0\ge 0,$$

$$\underset{t\to +\infty }{\lim }\frac{B(kt)}{A(t)}=0.$$

$$\rho (u;M)={\int }_{\Omega }M(u(x))dx<\infty (\text{resp.}{\int }_{\Omega }M(\lambda u(x))dx<\infty \text{for some}\lambda >0).$$

$$\|u\|_{(M)}=\inf\bigg{\{}\lambda>0\ :\ \int_{\Omega}M(\frac{u}{\lambda})\leq 1\bigg{\}},$$

$$\Vert u{\Vert }_{L^M(\Omega )}=\underset{\rho (v;\overline{M})\le 1}{\sup }{\int }_{\Omega }|u(x)||v(x)|dx.$$

$${\xi }_0(\rho )M(t)\le M(\rho t)\le {\xi }_1(\rho )M(t)\text{for}\rho ,t\ge 0$$

$${\xi }_0(\Vert u{\Vert }_{(M)})\le {\int }_{{\mathbb{R}}^N}M(|u|)dx\le {\xi }_1(\Vert u{\Vert }_{(M)})\text{for}u\in L^M({\mathbb{R}}^N).$$

$$W^{s,M}(\Omega)=\bigg{\{}u\in L^{M}(\Omega):\ \exists\lambda>0/\ \int_{\Omega}\int_{\Omega}M\bigg{(}\frac{\lambda(u(x)-u(y))}{|x-y|^{s}M^{-1}(|x-y|^{N})}\bigg{)}dxdy<\infty\bigg{\}}.$$

$$\Vert u{\Vert }_{(s,M)}=\Vert u{\Vert }_{(M)}+[u{]}_{(s,M)},$$

$$[u]_{(s,M)}=\inf\bigg{\{}\lambda>0:\ \int_{\Omega}\int_{\Omega}M\bigg{(}\frac{u(x)-u(y)}{\lambda|x-y|^{s}M^{-1}(|x-y|^{N})}\bigg{)}dxdy\leq 1\bigg{\}}.$$

$$\Vert u{\Vert }_{(M)}\le \mu [u{]}_{(s,M)},\forall u\in W_0^{s,M}(\Omega ).$$

$${\int }_0^1\frac{M^{-1}(\tau )}{{\tau }^{\frac{N+s}{N}}}d\tau <\infty$$

$${\int }_1^{+\infty }\frac{M^{-1}(\tau )}{{\tau }^{\frac{N+s}{N}}}d\tau =\infty .$$

Full text

[TABLE]

where $0<s<1$, $N\geq 2$, $(-\triangle)^{s}_{m}$ is fractional $M$-Laplace operator and the nonlinearity $f$ is sublinear as $|u|\rightarrow\infty$. The proof is based on the variant Fountain theorem established by Zou.

Keywords: Fractional Orlicz-Sobolev space, Compact embedding theorem, Fractional $M-$Laplacian, Fountain Theorem.

1 Introduction and main result

In this paper, we are concerned with the study of the nonlinear fractional $M$-Laplacian equation:

[TABLE]

where $0<s<1$, $N\geq 2$ and $M(t)=\displaystyle\int_{0}^{|t|}m(s)sds.$

In the last years, problem (1.1) has received a special attention for the case where $M(t)=\frac{1}{2}|t|^{2}$, that is, when it is of the form

[TABLE]

We do not intend to review the huge bibliography of equations like (1.2), we just emphasize that the most famous conditions on the potential $V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ are the following:

$(V_{1})$ $\quad\ \ \ V\in C(\mathbb{R}^{N},\mathbb{R})$ and $\inf_{x\in\mathbb{R}^{N}}V(x)\geq V_{0}>0$.

$(V_{2})$      There exists $\nu>0$ such that

[TABLE]

where $meas(.)$ denotes the Lebesgue measure in $\mathbb{R}^{N}$. We quote here [6, 26, 27] where the existence of infinitely many nontrivial solutions for the equation (1.2) have been obtained in connection with the geometry of the function $V$.

For the case where $s=1$, problem (1.1) becomes

[TABLE]

where the operator $\triangle_{m}u=div(m(|\nabla u|)\nabla u)$ named $M$-Laplacian. The reader can find more details involving this subject in [1, 9, 20, 21] and their references.

Notice that when $0<s<1$ and $M(t)=\frac{1}{p}|t|^{p}$ where $p>1$ the problem (1.1) gives back the fractional Schrödinger equation

[TABLE]

where $(-\triangle)^{s}_{p}$ is the non-local fractional $p$-Laplacian operator. Concerning the equation (1.3), in the last decade, many several existence and multiplicity results have been obtained by using different variational methods. In [21], the authors studied the existence of multiple ground state solutions for the problem (1.3), when the nonlinear term $f$ is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. Ambrosio [3] established an existence of infinitly solutions for the problem (1.3), when $f$ is $p$-superlinear and $V(x)$ can change sign. Moreover, fractional Schrödinger-type problems have been considered in some interesting papers [4, 16, 25]. The literature on non-local operators and on their applications is very interesting and, up to now, quite large. After the seminal papers by Caffarelli et al. [11, 12, 13], a large amount of papers were written on problems involving the fractional diffusion operator $(-\Delta)^{s}$ ($0<s<1$). We can quote [7, 14, 15, 23, 24] and the references therein. We also refer to the recent monographs [14, 22] for a thorough variational approach of non-local problems.

Contrary to the classical fractional Laplacian case that is widely investigated, the situation seems to be in a developing state when the new fractional $M$-Laplacian is present. In this context, the natural setting for studying problem (1.1) are fractional Orlicz-Sobolev spaces. Currently, as far as we know, the only results for fractional Orlicz-Sobolev spaces and fractional $M$-Laplacian operator are obtained in [2, 5, 10]. In particular, in [10], the authors define the fractional order Orlicz-Sobolev space associated to an $N$-function $M$ and a fractional parameter $0<s<1$ as

[TABLE]

The previous definition creates problems in the calculus and in the embedding results, for example, the Borel measure defined as $d\mu=\frac{dxdy}{|x-y|^{N}}$ is not finish in the neighbourhood of the origin, that’s whay, in [2], the authors introduced another definition of the fractional Orlicz-Sobolev space, i.e,

[TABLE]

The authors in [5] gave some further basic properties both on this function space and the related nonlocal operator.

Motivated by the above papers, under the suitable conditions $(V_{1})$ and $(V_{2})$ on the potential $V$ and exploiting the variant Fountain theorem, we aim to study the multiplicity of nontrivial weak solutions to (1.1) where the new fractional $M$-Laplacian is present. In this spirt, we deal with a new compact embedding theorem, also, we establish some useful inequalities which yields to apply the variational methods. As far as we know, all these results are new.

Related to functions $M$ and $f$, our hypotheses are the following:

Conditions on $m$ and $M$:

The function $m:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a $C^{1}$-function satisfying

1. ($m_{1}$)

$m(t)$, $(m(t)t)^{{}^{\prime}}>0$ for all $t>0$. 2. ($m_{2}$)

There exist $l,r\in]1,N[$ such that

[TABLE]

where

[TABLE]

Moreover, $m_{*}(t)t$ is such that the Sobolev conjugate function $M_{*}$ of $M$ is its primitive; that is,

[TABLE]

$(M_{1})$   There exists a positive constant $C$ such that

[TABLE]

where $M_{*}$ is the Sobolev conjugate of $M$.

$(M_{2})$ $\lim_{|t|\rightarrow+\infty}\displaystyle\frac{C|t|^{\mu}}{M(t)}=0,\ \forall 1<\mu\leq r.$

$(M_{3})$ The function $t\mapsto M(\sqrt{t}),\ t\in[0,\infty[$ is convex.

Conditions on $f$:

$(f_{1})$ $f(x,u)=p\xi(x)|u|^{p-2}u$, where $1<p<l$ is a constant and $\xi:\ \mathbb{R}^{N}\rightarrow\mathbb{R}$ is a positive continuous function such that $\xi\in L^{\frac{r}{r-p}}(\mathbb{R}^{N})$.

We mention some examples of functions $M$, whose function $m(t)$ satisfies the conditions $(m_{1})$-$(m_{2})$. The examples are the following:

$M(t)=|t|^{p}$ for $1<p<N$. 2. 2.

$M(t)=|t|^{p}+|t|^{q}$ for $1<p<q<N$ and $q\in]p,p^{*}[$ with $p^{*}=\frac{Np}{N-p}$. 3. 3.

$M(t)=(1+|t|^{2})^{\gamma}-1$ for $1<\gamma<\frac{N}{N-2}$.

Using the above hypotheses, we are able to state our main result.

Theorem 1.1**.**

Suppose that $(m_{1})-(m_{2})$, $(M_{1})-(M_{3})$, $(V_{1})-(V_{2})$ and $(f_{1})$ hold. Then, problem (1.1) possesses infinitely many nontrivial solutions.

This paper is organized as follows. In Section $2$, we give some definitions and fundamental properties of the spaces $L^{M}(\Omega)$ and $W^{s,M}(\Omega)$. In Section $3$, we prove some basic properties of the fractional Orlicz-Sobolev space and we show a compact embedding type theorem. Finally, in Section $4$, using a variant Fountain theorem, we prove our main result.

2 Preliminaries

In this preliminary section, for the reader’s convenience, we make a brief overview on the fractional Orlicz-Sobolev spaces studied in [2], and the associated fractional $M$-laplacian operator.

Let $M:\mathbb{R}\rightarrow\mathbb{R}_{+}$ be an $N$-function, i.e,

$M$ is even, continuous, convex, with $M(t)>t$ for $t>0$, 2. 2.

$\frac{M(t)}{t}\rightarrow 0$ as $t\rightarrow 0$ and $\frac{M(t)}{t}\rightarrow+\infty$ as $t\rightarrow+\infty$.

Equivalently, $M$ admits the representation:

[TABLE]

where $m:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is non-decreasing, right continuous, with $m(0)=0$, $m(t)>0\ \forall t>0$ and $m(t)\rightarrow\infty$ as $t\rightarrow\infty$. The conjugate $N$-function of $M$ is defined by

[TABLE]

where $\overline{m}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is given by $\overline{m}(t)=\sup\{s:\ m(s)\leq t\}$. Evidently we have

[TABLE]

which is known as the Young inequality. Equality holds in (2.4) if and only if either $t=m(s)$ or $s=\overline{m}(t)$.

In what follows, we say that an $N$-function $M$ verifies the $\triangle_{2}$ condition

[TABLE]

for some constant $K>0$. This condition can be rewritten in the following way: For each $s>0$, there exists $K_{s}>0$ such that

[TABLE]

If $A$ and $B$ are two $N$-functions, we say that $A$ is stronger than $B$ if

[TABLE]

for each $a>0$ and $x_{0}$ (depending on $a$), $B\prec\prec A$ in symbols. This is the case if and only if for every positive constante $k$

[TABLE]

The Orlicz class $K^{M}(\Omega)$ (resp. the Orlicz space $L^{M}(\Omega)$) is defined as the set of (equivalence classes of) real-valued measurable functions $u$ on $\Omega$ such that

[TABLE]

$L^{M}(\Omega)$ is a Banach space under the Luxemburg norm

[TABLE]

whose norm is equivalent to the Orlicz norm

[TABLE]

The next lemma and their proof can be found in [17].

Lemma 2.1**.**

Assume that $(m_{1})$ and $(m_{2})$ hold and let $\xi_{0}(t)=\min\{t^{l},t^{r}\}$, $\xi_{1}(t)=\max\{t^{l},t^{r}\}$, for all $t\geq 0$. Then,

[TABLE]

and

[TABLE]

Definition 2.2**.**

Let $M$ be an $N$-function. For a given domain $\Omega$ in $\mathbb{R}^{N}$ and $0<s<1$, we define the fractional Orlicz-Sobolev space $W^{s,M}(\Omega)$ as follows,

[TABLE]

This space is equipped with the norm,

[TABLE]

where $[.]_{(s,M)}$ is the Gagliardo semi-norm, defined by

[TABLE]

Let $W^{s,M}_{0}(\Omega)$ denote the closure of $C_{c}^{\infty}(\Omega)$ in the norm $\|.\|_{(s,M)}$ defined in (2.9).

Theorem 2.3**.**

[(Generalized Poincaré inequality)] Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ and let $s\in]0,1[$. Let $M$ be an $N$-function. Then there exists a positive constant $\mu$ such that,

[TABLE]

Therefore, if $\Omega$ is bounded and $M$ be an $N$-function, then $[u]_{(s,M)}$ is a norm of $W^{s,M}_{0}(\Omega)$ equivalent to $\|u\|_{(s,M)}$.

Let $M$ be a given $N$-function, satisfying the following conditions:

[TABLE]

and

[TABLE]

If (2.12) is satisfied, we define the inverse Sobolev conjugate $N$-function of $M$ as follows,

[TABLE]

Theorem 2.4**.**

Let $M$ be an $N$-function and $s\in]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}$ with $C^{0,1}$-regularity and bounded boundary. If (2.11) and (2.12) hold, then

[TABLE]

Moreover,

[TABLE]

is compact for all $B\prec\prec M_{*}$.

The fractional $M$-Laplacian operator is defined as

[TABLE]

where $P.V$ is the principal value.

This operator is well defined between $W^{s,M}(\mathbb{R}^{N})$ and its dual space $W^{-s,\overline{M}}(\mathbb{R}^{N})$. In fact, in [[2], lemma 3.5] the following representation formula is provided

[TABLE]

for all $v\in W^{s,M}(\mathbb{R}^{N})$.

3 Variational setting and some useful tools

In this section, we will first introduce the variational setting for problem (1.1). In view of the presence of potential $V(x)$, our working space is

[TABLE]

equipped with the following norm

[TABLE]

where

[TABLE]

We define the functional $G:E\rightarrow\mathbb{R}$ by

[TABLE]

where $h_{u}(x,y)=\displaystyle\frac{u(x)-u(y)}{|x-y|^{s}M^{-1}(|x-y|^{N})}.$

After integrating, we obtain from $(f_{1})$ that for any $(x,t)\in\mathbb{R}^{N}\times\mathbb{R}$

[TABLE]

In order to prove Theorem 1.1, we will consider the following family of functionals

[TABLE]

with $\lambda\in[1,2]$, $u\in E$ and

[TABLE]

We will show that $I_{\lambda}$ satisfies the assumptions of the following variant of fountain Theorem due to Zou [28].

Theorem 3.1**.**

Let $(E,\|\|)$ be a Banach space and $E=\overline{\bigoplus_{j\in\mathbb{N}}X_{j}}$ with $dim\ X_{j}<\infty$ for any $j\in\mathbb{N}$. Set $Y_{k}=\bigoplus_{j=1}^{k}X_{j}$ and $Z_{k}=\bigoplus_{j=k}^{\infty}X_{j}$. Consider the following $C^{1}$-functional $I_{\lambda}:\ E\rightarrow\mathbb{R}$ defined by

[TABLE]

Assume that $I_{\lambda}$ satisfies the following assumptions:

$(i)$* $I_{\lambda}$ maps bounded sets to bounded sets for $\lambda\in[1,2]$ and $I_{\lambda}(-u)=I_{\lambda}(u)$ for all $(\lambda,u)\in[1,2]\times E$.*

$(ii)$* $B(u)\geq 0$, $B(u)\rightarrow+\infty$ as $\|u\|\rightarrow+\infty$ on any finite dimensional subspace of $E$.*

$(iii)$* There exists $r_{k}>\rho_{k}$ such that*

[TABLE]

and

[TABLE]

Then there exist $\lambda_{n}\rightarrow 1$, $u_{\lambda_{n}}\in Y_{n}$ such that

[TABLE]

Particularly, if $(u_{\lambda_{n}})$ has a convergent subsequence for every $k$, then $I_{1}$ has infinitely many nontrivial critical points $\{u_{k}\}\in E\backslash\{0\}$ satisfying $I_{1}(u_{k})\rightarrow 0^{-}$ as $k\rightarrow\infty$.

Now we give the definition of weak solution for the problem (1.1). We define the functional $I_{1}$ on $E$ by

[TABLE]

where

[TABLE]

Definition 3.2**.**

We say that $u\in E$ is a weak solution to $\eqref{eq}$ if $u$ satisfies

[TABLE]

for all $v\in E$.

The functional $I_{\lambda}$ is well defined on $E$ moreover $I_{\lambda}\in C^{1}(E,\mathbb{R})$ and

[TABLE]

Then the critical points of $I_{1}$ are weak solutions to (1.1).

Now, we introduce some important inequalities that show that the functional $I_{\lambda}$ satisfies the hypothesis of Theorem 3.1.

Lemma 3.3**.**

*we assume that $(m_{1})$, $(m_{2})$ and $(V_{1})$ are satisfied. Then, the following properties hold true:

$(i)$* *$\xi_{0}([u]_{(s,M)})\leq G(u)\leq\xi_{1}([u]_{(s,M)})\ \forall u\in E,$

$(ii)$* $\xi_{0}(\|u\|_{(V,M)})\leq\displaystyle\int_{\mathbb{R}^{N}}V(x)M(u)dx\leq\xi_{1}(\|u\|_{(V,M)})\ \forall u\in E,$*

Proof.

$(i)$ By Lemma 2.1, we Know that

[TABLE]

It follows that

[TABLE]

Having in mind that, $\|h_{u}\|_{L^{M}(\mathbb{R}^{2N})}=[u]_{(s,M)},$ we obtain

[TABLE]

$(ii)$ Using Lemma 2.1 and Choosing $\rho=\|u\|_{(V,M)}$, we have

[TABLE]

then,

[TABLE]

From the definition of the norm (2.7), we obtain,

[TABLE]

Using the similar reasoning with $\rho=\|u\|_{(V,M)}-\epsilon$ and $\epsilon>0$, we get

[TABLE]

then

[TABLE]

Letting $\epsilon\rightarrow 0$ in the above inequality, we obtain

[TABLE]

The proof of Lemma 3.3 is complete. ∎

Now we show that the following compactness result holds.

Lemma 3.4**.**

We suppose that $(m_{1})$ and $(m_{2})$ are satisfied. Let $\Phi$ be an $N$-function satisfying the $\triangle_{2}$ condition, $\Phi\prec\prec M_{*}$ and

[TABLE]

Under the assumption $(V_{1})$ and $(V_{2})$, the embedding from $E$ into $L^{\Phi}(\mathbb{R}^{N})$ is compact.

Proof.

Let $(u_{n})\subset E$ be a sequence verifying $u_{n}\rightharpoonup 0\ \text{in}\ E.$ We have to show that $u_{n}\rightarrow 0$ in $L^{\Phi}(\mathbb{R}^{N})$. By using Theorem 1.1 we know that $u_{n}\rightarrow 0$ in $L^{\Phi}_{loc}(\mathbb{R}^{N})$. Thus it suffices to show that, for any $\epsilon>0$, there exists $R>0$ such that

[TABLE]

here $B^{c}_{R}(0)=\mathbb{R}^{N}\setminus B_{R}(0)$.

Choose $(y_{i})_{i\in\mathbb{N}}\subset\mathbb{R}^{N}$ such that $\mathbb{R}^{N}\subset\bigcup_{i\in\mathbb{N}}B_{\nu}(y_{i})$ and each point $x\in\mathbb{R}^{N}$ is contained in at most $2^{N}$ such balls $B_{\nu}(y_{i})$. Let

[TABLE]

The fact that $(u_{n})$ converges weakly to $u$ in $E$ implies that $\|u_{n}\|_{E}\leq T,\ \forall n\in\mathbb{N}$ with $T>0$. From the $\triangle_{2}$ condition, there is $K>0$ such that

[TABLE]

Given $\epsilon>0$, by (3.21), there is $R>0$ such that

[TABLE]

Combining (3.22) and (3.23), we get

[TABLE]

and this can be made arbitrarily small by choosing $L$ large.

Take $C$ an $N$-function such that $C\circ\Phi\prec\prec M$ and let $\overline{C}$ be the conjugate of $C$. By Theorem $4.17.4$ in [18], there exist $K^{{}^{\prime}}>0$ such that

[TABLE]

Claim1: \xi_{1}(\|u_{n}\|_{(M)})\leq\xi_{1}\bigg{(}\displaystyle\frac{1}{V_{0}}\xi_{1}(\|u_{n}\|_{E})+1\bigg{)}

Indeed, using Lemma 3.3, we get

[TABLE]

We fixe $L>0$. Combining (3.25), claim $1$ and Lemma 2.1 and applying the Hölder inequality, we infer that

[TABLE]

where $\epsilon_{R}=\sup_{y_{i}}\|\chi_{B_{R,L}\bigcap B_{\nu}(y_{i})}\|_{L^{\overline{C}}(B_{\nu}(y_{i}))}$ and $K^{{}^{\prime\prime}}>0$. By assumption $(V_{2})$ and Proposition 4.6.9 in [18] we can infer that $\epsilon_{R}\rightarrow 0$ as $R\rightarrow\infty$. Thus we may make this term small by choosing $R$ large. Combining (3.24) and (3.26) we get our desired result. ∎

Corollary 3.5**.**

Under $(M_{1})$ and $(M_{2})$, the embedding from $E$ into $L^{\mu}(\mathbb{R}^{N})$ is compact for all $1<\mu\leq r$.

Proof.

Let $\Phi(t)=C|t|^{\mu}$. By condition $(M_{1})$, $(M_{2})$ and applying Lemma 3.4, we can deduce that $E$ is compactly embedded in $L^{\mu}(\mathbb{R}^{N})$ for all $1<\mu\leq r$. ∎

Lemma 3.6**.**

The functional $A$ is weakly lower semi-continuous.

Proof.

By Lemma $3.3$ in [5], it is enough to show that $\Psi$ is weakly lower semi-continuous. Let $(u_{n})\subset E$ be a sequence which converges weakly to $u$ in $E$. Since $E$ is compactly embedded in $L^{r}(\mathbb{R}^{N})$ it follows that $(u_{n})$ converges strongly to $u$ in $L^{r}(\mathbb{R}^{N})$. Then, up to a subsequence, we obtain

[TABLE]

This along with Fatou’s lemma yield

[TABLE]

Therefore, $A$ is weakly lower semi-continuous. The proof of Lemma 3.6 is complete. ∎

Lemma 3.7**.**

If $u_{n}\rightharpoonup u$ in $E$ and

[TABLE]

then $u_{n}\rightarrow u$ in $E$.

Proof.

Since $(u_{n})$ converges weakly to $u$ in $E$ implies that $([u_{n}]_{(s,M)})$ and $(\|u_{n}\|_{(V,M)})$ are a bounded sequences of real numbers. That fact and relations $(i)$ and $(ii)$ from lemma 3.3 imply that the sequences $(G(u_{n}))$ and $(\Psi(u_{n}))$ are bounded, it means that the sequence $(A(u_{n}))$ is bounded. Then, up to a subsequence, we deduce that $A(u_{n})\rightarrow c$. Furthermore, Lemma 3.6, implies

[TABLE]

On the other hand, since $A$ is convex, we have

[TABLE]

Therefore, combinings (3.28) and (3.29) and the hypothesis (3.27), we conclude that $A(u)=c$.

Taking into account that $\displaystyle\frac{u_{n}+u}{2}$ converges weakly to $u$ in $E$ and using again the weak lower semi-continuity of $A$ we find

[TABLE]

We assume by contradiction that $(u_{n})$ does not converge to $u$ in $E$. Then by $(i)$ and $(ii)$ in lemma 3.3 it follows that there exist $\epsilon>0$ and a subsequence $(u_{n_{m}})$ of $(u_{n})$ such that

[TABLE]

On the other hand, relations (2.5) and $(M_{3})$ enable us to apply [[19], theorem 2.1] in order to obtain

[TABLE]

Letting $m\rightarrow\infty$ in the above inequality we obtain

[TABLE]

and that is a contradiction with (3.30). It follows that $(u_{n})$ converges strongly to $u$ in $E$ and lemma 3.7 is proved. ∎

4 Proof of Theorem 1.1

We further need the following lemmas.

Lemma 4.1**.**

Let $(V_{1})$, $(V_{2})$, $(M_{1})$, $(M_{2})$ and $(f_{1})$ be satisfied. Then $B(u)\geq 0$. Furthermore, $B(u)\rightarrow\infty$ as $\|u\|\rightarrow\infty$ on any finite dimensional subspace of $E$.

Proof.

Evidently $B(u)\geq 0$ follows by $(f_{1})$. We claim that for any finite dimensional subspace $F\subset E$ there exists a constant $\epsilon>0$ such that

[TABLE]

We argue by contradiction and we suppose that for any $n\in\mathbb{N}$ there exists $0\neq u_{n}\in F$ such that

[TABLE]

For each $n\in\mathbb{N}$, let $v_{n}=\displaystyle\frac{u_{n}}{\|u_{n}\|}\in F$. Then $\|v_{n}\|=1$ for all $n\in\mathbb{N}$, and

[TABLE]

Up to a subsequence, we may assume that $v_{n}\rightarrow v$ in $E$ for some $v\in F$ since $F$ is a finite dimensional space. Clearly $\|v\|=1$. Consequently, there exists a constant $\delta_{0}>0$ such that

[TABLE]

In fact, if not, then we have

[TABLE]

which implies that

[TABLE]

This together $(f_{1})$ yields $v=0$, which is in contradiction to $\|v\|=1$.

By using Corollary 3.5 and the fact that all norms are equivalent on $F$, we deduce that

[TABLE]

By the Hölder inequality, it holds that

[TABLE]

Set

[TABLE]

and for all $n\in\mathbb{N}$,

[TABLE]

Taking into account (4.35) and (4.36), we get

[TABLE]

for $n$ large enough. Therefore we obtain

[TABLE]

which contradicts (4.37). For the $\epsilon$ given in (4.34), let

[TABLE]

Then by (4.34),

[TABLE]

Therefore

[TABLE]

This implies that $B(u)\rightarrow\infty$ as $\|u\|\rightarrow\infty$ on any finite dimensional subspace of $E$. The proof is complete. ∎

Lemma 4.2**.**

Assume that $(m_{1})-(m_{2})$, $(M_{1})-(M_{2})$, $(V_{1})-(V_{2})$ and $(f_{1})$ are satisfied. Then there exists a sequence $\rho_{k}\rightarrow 0^{+}$ as $k\rightarrow\infty$ such that

[TABLE]

and

[TABLE]

Proof.

Using Lemma 3.3, for any $u\in Z_{k}$ and $\lambda\in[1,2]$, we can see that

[TABLE]

Let

[TABLE]

By the next lemma, there hold

[TABLE]

Combining (4.38) and (4.39), we have

[TABLE]

For each $k\in\mathbb{N}$, choose

[TABLE]

Since $1<p<r$, then by (4.40), we have

[TABLE]

and so, for $k$ large enough, we have $\rho_{k}\leq 1$. Then, by Lemma 2.1,

[TABLE]

By (4.41), (4.42) and (4.44), direct computation shows

[TABLE]

Besides, by (4.41), for each $k\in\mathbb{N}$, we have

[TABLE]

for all $\lambda\in[1,2]$ and $u\in Z_{k}$ with $\|u\|\leq\rho_{k}$. Therefore,

[TABLE]

Combining (4.40) and (4.43), we have

[TABLE]

The proof is complete. ∎

Lemma 4.3**.**

We have that

[TABLE]

Proof.

It is clear that $l_{k}$ is decreasing with respect to $k$ so there exist $l\geq 0$ such that $l_{k}\rightarrow 0$ as $k\rightarrow+\infty$. For any $k\geq 0$, there exists $u_{k}\in Z_{k}$ such that $\|u_{k}\|=1$ and $\|u_{k}\|_{L^{r}(\mathbb{R}^{N})}\geq\frac{l_{k}}{2}$. By definition of $Z_{k}$, $u_{k}\rightharpoonup 0$ in $E$. Lemma 3.4 implies that $u_{k}\rightarrow 0$ in $L^{r}(\mathbb{R}^{N})$. Thus we proved that $l=0$. ∎

Lemma 4.4**.**

Under the hypotheses and the sequence $\rho_{k}$ of in Lemma 4.2, there exists $r_{k}>\rho_{k}$ for any $k\in\mathbb{N}$ such that

[TABLE]

Proof.

By using the fact that $Y_{k}$ is with finite dimensional and (4.34), we can find $\epsilon_{k}>0$ such that

[TABLE]

where $\Lambda_{u}^{k}:=\{x\in\mathbb{R}^{N}:\ \xi(x)|u(x)|^{p}\geq\epsilon_{k}\|u\|^{p}\}$. By (4.45), for any $k\in\mathbb{N}$, we have

[TABLE]

for all $u\in Y_{k}$ with $\|u\|\leq\min\{\rho_{k},2^{-\frac{1}{l-p}}\epsilon_{k}^{\frac{2}{l-p}},1\}$. If we choose

[TABLE]

and using (4.46), we deduce that

[TABLE]

∎

Proof of Theorem 1.1:.

From Lemma 3.4 we can see that $I_{\lambda}$ maps bounded sets to bounded sets uniformly for $\lambda\in[1,2]$. Moreover, $I_{\lambda}$ is even. Then the condition $(i)$ in Theorem 3.1 is satisfied. Besides, Lemma 4.1 shows that the condition $(ii)$ holds while Lemma 4.2 together with Lemma 4.4 implies that the condition $(iii)$ holds. Therefore, by Theorem 3.1, for each $k\in\mathbb{N}$, there exist $\lambda_{n}\rightarrow 1,$ $u_{\lambda_{n}}\in Y_{n}$ such that

[TABLE]

Claim 2: The sequence $(u_{\lambda_{n}})_{n\in\mathbb{N}}$ obtained in (4.47) is bounded in $E$.

For the sake of notational simplicity, in what follows we always set $u_{n}=u_{\lambda_{n}}$ for all $n\in\mathbb{N}$.

In fact, combining (4.47), Lemmas 3.3 and 3.4 and the Hölder inequality, we obtain

[TABLE]

for some $C_{0}>0$ and $\tau>0$, where $\theta=r$ or $\theta=l$. Therefore, the claim above is true since $p<\theta$.

Claim 3: The sequence $(u_{n})$ has a strong convergent subsequence for every $k$.

In view of Claim $2$, without loss of generality, we may assume

[TABLE]

for some $u_{0}\in E$. Hence

[TABLE]

Therefore, by Lemma 3.7, we can deduce that $u_{n}$ converges strongly to $u_{0}$ in $E$.

Now from the last assertion of Theorem 3.1, we know that $I_{1}$ has infinitely many nontrivial critical points. Therefore, (1.1) possesses infinitely many nontrivial solutions. The proof of Theorem 1.1 is complete. ∎