Multiple positive solutions for a Schr\"{o}dinger logarithmic equation

TL;DR

This paper proves the existence of multiple positive solutions for a logarithmic Schrödinger equation in  space, showing that the potential's shape influences the number of solutions for small psilon.

Contribution

It demonstrates how the shape of the potential function affects the number of solutions in a logarithmic Schrd6dinger equation using variational methods.

Findings

Multiple positive solutions exist for small psilon.

The shape of the potential function influences the number of solutions.

Variational methods are effective in establishing solution multiplicity.

Abstract

This article concerns with the existence of multiple positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $\epsilon >0$, $N \geq 1$ and $V$ is a continuous function with a global minimum. Using variational method, we prove that for small enough $\epsilon>0$, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions.

Equations

$$\left\{\begin{array}[]{lc}-{\epsilon}^{2}\Delta u+V(x)u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),&\\ \end{array}\right.$$

$$iϵ{\partial }_t\Psi =-{ϵ}^2\Delta \Psi +(W(x)+w)\Psi -\Psi \log |\Psi {|}^2,\Psi :[0,\infty )\times {\mathbb{R}}^N\to \mathbb{C},N\ge 1,$$

$$-{ϵ}^2\Delta u+V(x)u=u\log u^2,\text{mbox}in{\mathbb{R}}^N.$$

$${\hat{I}}_{ϵ}(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}({ϵ}^2|\nabla u{|}^2+V(x)|u{|}^2)dx-{\int }_{{\mathbb{R}}^N}H(u)dx,$$

$$H(t)={\int }_0^{t}s\log s^{2}ds=\frac{-t^2}{2}+\frac{t^2\log t^2}{2},\forall t\in \mathbb{R}$$

$${\hat{I}}_{ϵ}(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}({ϵ}^2|\nabla u{|}^2+(V(x)+1)|u{|}^2)dx-\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^2\log u^{2}dx.$$

$$\left\{\begin{array}[]{lc}-\Delta u+u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}).&\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lc}(-\Delta)^{s}u+u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{s}(\mathbb{R}^{N}),&\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lc}-\Delta u+V(x)u=Q(x)u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),&\\ \end{array}\right.$$

$$\underset{x\in {\mathbb{R}}^N}{\min }Q(x)>0\text{mbox}and\underset{x\in {\mathbb{R}}^N}{\min }(V+Q)(x)>0.$$

$$\left\{\begin{array}[]{lc}-\Delta u+V(x)u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),&\\ \end{array}\right.$$

$$\underset{|x|\to +\infty }{\lim }V(x)=V_{\infty }\text{mbox}where{V}_{\infty }+1\in (0,+\infty ].$$

$$\left\{\begin{array}[]{lc}-{\epsilon}^{2}\Delta u+V(x)u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),&\\ \end{array}\right.$$

$$\underset{|x|\to \infty }{\lim }V(x)=V_{\infty },$$

$$1=V(z_i)=\underset{x\in {\mathbb{R}}^N}{\min }V(x),\mathrm{for}1\le i\le l.$$

$$\left\{\begin{array}[]{lc}-\Delta u+V(\epsilon x)u=u\log u^{2},&\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}).&\\ \end{array}\right.$$

$${\int }_{{\mathbb{R}}^N}(\nabla u\cdot \nabla v+V(ϵx)u\cdot v)dx={\int }_{{\mathbb{R}}^N}uv\log u^{2}dx,\text{mbox}forallv\in C_0^{\infty }({\mathbb{R}}^N).$$

$$\{\begin{array}{rl}& -\Delta u+u=A(ϵx)|u{|}^{p-2}u\mathrm{in}{\mathbb{R}}^N,\\ & u\in H^1({\mathbb{R}}^N).\end{array}$$

$$g^{+}(x)=\max \{g(x),0\}\text{mbox}and{g}^{-}(x)=\max \{-g(x),0\}.$$

$$\left\{\begin{array}[]{lc}-\Delta u+V(0)u=u\log u^{2},&\quad\mbox{in}\quad\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}).&\\ \end{array}\right.$$

$$J_{0}(u)=\displaystyle\frac{1}{2}\int\big{(}|\nabla u|^{2}+({V(0)}+1)|u|^{2}\big{)}dx-\displaystyle\frac{1}{2}\int u^{2}\log u^{2}dx.$$

$$c_0:=\underset{u\in {\mathcal{N}}_0}{\inf }{J}_0(u),$$

$${\mathcal{N}}_0=\left\{u\in D(J_0)\backslash \{0\}:J_0(u)=\frac{1}{2}\int |u{|}^{2}dx\right\}$$

$$D(J_0)=\left\{u\in H^1({\mathbb{R}}^N):J_0(u)<+\infty \right\}.$$

$$c_{\infty }=\underset{u\in {\mathcal{N}}_{\infty }}{\inf }{J}_{\infty }(u),$$

$$c_0<c_{\infty }.$$

$$F_{1}(s)=\left\{\begin{array}[]{lc}0,&\;s=0\\ -\frac{1}{2}s^{2}\log s^{2},&\;0<|s|<\delta\\ -\frac{1}{2}s^{2}(\log\delta^{2}+3)+2\delta|s|-\frac{1}{2}\delta^{2},&\;|s|\geq\delta\end{array}\right.$$

$$F_{2}(s)=\left\{\begin{array}[]{lc}0,&\;|s|<\delta\\ \frac{1}{2}s^{2}\log(s^{2}/\delta^{2})+2\delta|s|-\frac{3}{2}s^{2}-\frac{1}{2}\delta^{2},&\;|s|\geq\delta.\end{array}\right.$$

$$F_2(s)-F_1(s)=\frac{1}{2}{s}^2\log s^2,\forall s\in \mathbb{R},$$

$$J_0(u)={\Phi }_0(u)+\Psi (u)\text{mbox}and{J}_{\infty }(u)={\Phi }_{\infty }(u)+\Psi (u),u\in H^1({\mathbb{R}}^N)$$

Full text

[TABLE]

where $\epsilon>0$, $N\geq 1$ and $V$ is a continuous function with a global minimum. Using variational method, we prove that for small enough $\epsilon>0$, the ”shape” of the graph of the function $V$ affects the number of nontrivial solutions.

2010 Mathematics Subject Classification: 35A15, 35J10; 35B09

Keywords: Variational method, Logarithmic Schrödinger equation, Positive solutions, Multiple solutions.

1 Introduction

Recently, the logarithmic Schrödinger equation given by

[TABLE]

has received a special attention because it appears in a lot of physical applications, such as quantum mechanics, quantum optics, nuclear physics, transport and diffusion phenomena, open quantum systems, effective quantum gravity, theory of superfluidity and Bose-Einstein condensation (see [14] and the references therein). In its turn, standing waves solution, $\Psi$, for this logarithmic Schrödinger equation is related to solutions of the equation

[TABLE]

Besides the importance in applications, this last equation is very interesting in the mathematical point of view, because it arises a lot of difficulties to apply variational methods in order to find a solution for it. The natural candidate for the associated energy functional would formally be the functional

[TABLE]

where

[TABLE]

that is,

[TABLE]

However, this functional is not well defined in $H^{1}(\mathbb{R}^{N})$ because there is $u\in H^{1}(\mathbb{R}^{N})$ such that $\int_{\mathbb{R}^{N}}u^{2}\log u^{2}\,dx=-\infty$. In order to overcome this technical difficulty some authors have used different techniques.

In [3], d’Avenia, Montefusco and Squassina have studied the existence of multiple solutions for a logarithmic elliptic equation of the type

[TABLE]

The authors obtained solutions for this equation by applying the non-smooth critical point theory, found in Degiovanni and Zani [5], to the energy functional defined on the space of radial functions $H_{rad}^{1}(\mathbb{R}^{N})$. In [4], d’Avenia, Squassina and Zenari have used the same approach to show the existence of solution for a fractional logarithmic Schrödinger equation of the type

[TABLE]

for $s\in(0,1)$ and $N>2s$.

In [10], Squassina and Szulkin have showed the existence of multiple solutions for the following class of problem

[TABLE]

where $V,Q:\mathbb{R}^{N}\to\mathbb{R}$ are 1-periodic continuous functions verifying

[TABLE]

In that paper, the authors have used the minimax principles for lower semicontinuous functionals, developed by Szulkin [12], to prove the existence of geometrically distinct multiple solutions and the existence of a ground state solution. The multiple solutions follow by genus theory found in [12], while the existence of ground state follows by a specific deformation lemma, see [10, Lemma 2.14].

Later, Ji and Szulkin in [6] have established the existence of multiple solutions for a problem of the type

[TABLE]

where $V:\mathbb{R}^{N}\to\mathbb{R}$ is a continuous function that satisfies

[TABLE]

With the same approach explored in [10], the above authors showed that if $V_{\infty}=+\infty$, then $(P_{4})$ has infinitely many solutions. If $V_{\infty}\in(-1,+\infty)$ and the spectrum $\sigma(-\Delta+V+1)\subset(0,+\infty)$, then problem $(P_{4})$ has a ground state solution. In [9], Tanaka and Zhang have studied the existence of solution for $(P_{3})$. In that very nice paper the authors have observed that the positivity of $V$ is not essential.

Finally, in a recent work, Alves and de Morais Filho in [1] have used the minimax method found in [12] to show the existence and concentration of positive solution for the problem

[TABLE]

where $\epsilon>0,N\geq 1$ and $V$ is a continuous function with a global minimum.

Motivated by results found in the above mentioned papers, in the present paper we intend to study the existence of multiple solutions for problem (1.2) by supposing the following conditions on potential $V$:

• ($V1$)

$V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a continuous function such that

[TABLE]

with $0<V(x)<V_{\infty}$ for any $x\in\mathbb{R}^{N}$.

• ($V2$)

There exist $l$ points $z_{1},z_{2},...,z_{l}$ in $\mathbb{R}^{N}$ with $z_{1}=0$ such that

[TABLE]

By a change of variable, we know that problem (1.2) is equivalent to the problem

[TABLE]

Definition 1.1**.**

For us, a positive solution of (1.3) means a positive function $u\in H^{1}(\mathbb{R}^{N})\setminus\{0\}$ such that $u^{2}\log u^{2}\in L^{1}(\mathbb{R}^{N})$ and

[TABLE]

The main result to be proved is the theorem below.

Theorem 1.1**.**

Suppose that $V$ satisfies $(V1)$ and ($V2$). Then there is $\epsilon_{*}>0$ such that problem (1.2) has at least $\l$ positive solutions in $H^{1}(\mathbb{R}^{N})$ for all $\epsilon\in(0,\epsilon_{*})$.

In the proof of Theorem 1.1, we adapt for our problem some ideas explored in Cao and Noussair [2], where the existence and multiplicity of solutions have considered for the following class of problem

[TABLE]

Using Ekeland’s variational principle and concentration compactness principle of Lions [7], Cao and Noussair proved that if $A$ has $l$ equal maximum points, then problem (1.5) has at least $l$ positive solutions and $l$ nodal solutions if $\epsilon>0$ is small enough. We would like to point out that different of [2], where the energy functional is $C^{1}$, we cannot work directly with the energy functional associated with (1.3) because it is not continuous, and so, it is not $C^{1}$. Have this in mind, for each $R>0$, we first find a solution $u_{\epsilon,R}\in H_{0}^{1}(B_{R}(0))$, and after, taking the limit of $R\to+\infty$ we get a solution for the original problem.

The plan of the paper is as follows. In Section 2 we show some preliminary results which will be used later on. In Section 3 we prove the existence of multiple solutions for an auxiliary problem, while in Section 4 we prove Theorem 1.1.

Notation: From now on in this paper, otherwise mentioned, we use the following notations:

• •

$B_{r}(u)$ is an open ball centered at $u$ with radius $r>0$, $B_{r}=B_{r}(0)$.

• •

If $g$ is a mensurable function, the integral $\displaystyle\int_{\mathbb{R}^{N}}g(x)\,dx$ will be denoted by $\displaystyle\int g(x)\,dx$. Moreover, we denote by $g^{+}$ and $g^{-}$ the positive and negative part of $g$ given by

[TABLE]

• •

$C$ denotes any positive constant, whose value is not relevant.

• •

$|\,\,\,|_{p}$ denotes the usual norm of the Lebesgue space $L^{p}(\mathbb{R}^{N})$, for $p\in[1,+\infty]$.

• •

$H_{c}^{1}(\mathbb{R}^{N})=\{u\in H^{1}(\mathbb{R}^{N})\,:\,u\,\,\mbox{has compact support}\,\}.$

• •

$o_{n}(1)$ denotes a real sequence with $o_{n}(1)\to 0$ as $n\to+\infty$.

• •

$2^{*}=\frac{2N}{N-2}$ if $N\geq 3$ and $2^{*}=+\infty$ if $N=1,2$.

2 Preliminaries

Hereafter, we consider the problem

[TABLE]

The corresponding energy functional associated to (2.1) will be denoted by $J_{0}:H^{1}(\mathbb{R}^{N})\rightarrow(-\infty,+\infty]$ and defined as

[TABLE]

In [10] is proved that problem (2.1) has a positive solution attained at the infimum

[TABLE]

where

[TABLE]

and

[TABLE]

Mutatis mutandis the previous notations, we shall also use the energy level

[TABLE]

corresponding to problem (2.1), replacing $V(0)$ by $V_{\infty}$. Using the definition of $c_{0}$ and $c_{\infty}$, it follows that

[TABLE]

Related to the numbers $c_{0}$ and $c_{\infty}$, we would like to point out that they are the mountain pass levels of the functionals $J_{0}$ and $J_{\infty}$ respectively.

Following the approach explored in [1, 6, 10], due to the lack of smoothness of $J_{0}$ and $J_{\infty}$, let us decompose them into a sum of a $C^{1}$ functional plus a convex lower semicontinuous functional, respectively. For $\delta>0$, let us define the following functions:

[TABLE]

and

[TABLE]

Therefore

[TABLE]

and the functionals $J_{0},J_{\infty}:H^{1}(\mathbb{R}^{N})\rightarrow(-\infty,+\infty]$ may be rewritten as

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

It was proved in [6] and [10] that $F_{1}$ and $F_{2}$ verify the following properties:

[TABLE]

If $\delta>0$ is small enough, $F_{1}$ is convex, even, $F_{1}(s)\geq 0$ for all $s\in\mathbb{R}$ and

[TABLE]

For each fixed $p\in(2,2^{*})$, there is $C>0$ such that

[TABLE]

3 An auxiliary functional

In what follows, let us fix $R_{0}>0$ such that $z_{i}\in B_{R_{0}}(0)$ for all $i\in\{0,1,...,\l\}$. Moreover, for each $R>R_{0}$, we set the functional

[TABLE]

It is easy to check that $J_{\epsilon,R}\in C^{1}(H_{0}^{1}(B_{R}(0)),\mathbb{R})$ with

[TABLE]

Hereafter, $H_{0}^{1}(B_{R}(0))$ is endowed with the norm

[TABLE]

which is also a norm in $H^{1}(\mathbb{R}^{N})$. Moreover, this norm is equivalent to the usual norms in $H_{0}^{1}(B_{R}(0))$ and $H^{1}(\mathbb{R}^{N})$ respectively.

In the sequel, $\mathcal{N}_{\epsilon,R}$ denotes the Nehari manifold associated with $J_{\epsilon,R}$, that is,

[TABLE]

or equivalently,

[TABLE]

The next three lemmas show that $J_{\epsilon,R}$ verifies the mountain pass geometry and the well known $(PS)$ condition.

Lemma 3.1**.**

For all $\epsilon>0,R>R_{0}$, the functional $J_{\epsilon,R}$ has the mountain pass geometry.

Proof.

$(i)$: Note that $J_{\epsilon,R}(u)\geq\displaystyle\frac{1}{2}\|u\|_{\epsilon}^{2}-\int_{B_{R}(0)}F_{2}(u)dx$. Hence, from (2.11), fixed $p\in(2,2^{*})$, it follows that

[TABLE]

for some $C_{1}>0$ and $\|u\|_{\epsilon}>0$ small enough. Here the constant $C_{1}$ does not depend on $\epsilon$ and $R>0$.

$(ii)$: Let us fix $u\in D(J_{\epsilon,R})\setminus\{0\}$ with $suppu\subset B_{R_{0}}(0)$ and $s>0$. Using (2.4) we get

[TABLE]

Thereby, there is $s_{0}>0$ independent of $\epsilon>0$ small enough and $R>R_{0}$ such that $J_{\epsilon,R}(s_{0}u)<0$. ∎

Lemma 3.2**.**

All (PS) sequences of $J_{\epsilon,R}$ are bounded in $H_{0}^{1}(B_{R}(0))$.

Proof.

Let $(u_{n})\subset H_{0}^{1}(B_{R}(0))$ be a $(PS)_{d}$ sequence. Then,

[TABLE]

for some $C>0$. Consequently

[TABLE]

Now, let us employ the following logarithmic Sobolev inequality found in [8],

[TABLE]

for all $a>0$. Fixing $\frac{a^{2}}{\pi}=\frac{1}{4}$ and $\xi\in(0,1)$, the inequalities (3.3) and (3.4) yield

[TABLE]

Since

[TABLE]

assertion (3.5) assures that

[TABLE]

showing that the sequence $(u_{n})$ is bounded in $H_{0}^{1}(B_{R}(0))$.

∎

Lemma 3.3**.**

The mountain pass level $c_{\epsilon,R}$ of $J_{\epsilon,R}$ can be characterized by

[TABLE]

Proof.

See [1, Lemma 3.3].

∎

Lemma 3.4**.**

The functional $J_{\epsilon,R}$ satisfies the $(PS)$ condition.

Proof.

Let $(u_{n})\subset H_{0}^{1}(B_{R}(0))$ be a $(PS)_{d}$ sequence for $J_{\epsilon,R}$, that is,

[TABLE]

By Lemma 3.2, we can assume that there is $u\in H_{0}^{1}(B_{R}(0))$ and a subsequence of $(u_{n})$, still denoted by itself, such that

[TABLE]

[TABLE]

and

[TABLE]

Setting $f(t)=t\log t^{2}$, $F(t)=\int_{0}^{t}f(s)\,ds=\frac{1}{2}(t^{2}\log t^{2}-t^{2})$ for all $t\in\mathbb{R}$ and fixing $p\in(2,2^{*})$, there is $C>0$ such that

[TABLE]

and

[TABLE]

Hence, by the Sobolev embeddings,

[TABLE]

and

[TABLE]

Now, using the limits $J^{\prime}_{\epsilon,R}(u_{n})u_{n}=J^{\prime}_{\epsilon,R}(u_{n})u+o_{n}(1)$, it is easy to see that

[TABLE]

showing the lemma. ∎

In the following, let us fix $\rho_{0}>0$ satisfying $\overline{B_{\rho_{0}}(z_{i})}\cap\overline{B_{\rho_{0}}(z_{j})}={\O}$ for $i\neq j$ and $i,j\in\{1,...,l\}$, $\bigcup^{l}_{i=1}B_{\rho_{0}}(z_{i})\subset B_{R_{0}}(0)$ and $K_{\frac{\rho_{0}}{2}}=\bigcup^{l}_{i=1}\overline{B_{\frac{\rho_{0}}{2}}(z_{i})}$. Moreover, we also set the function $Q_{\epsilon}:H^{1}(\mathbb{R}^{N})\setminus\{0\}\rightarrow\mathbb{R}^{N}$ by

[TABLE]

where $\chi:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ is given by

[TABLE]

and $g:\mathbb{R}^{N}\to\mathbb{R}$ is a radial positive continuous function such that

[TABLE]

The following lemma is very useful to obtain $(PS)_{c}$ sequences associated with $J_{\epsilon,R}$.

Lemma 3.5**.**

There exist $\alpha_{0}>0$, $\epsilon_{1}\in(0,\epsilon_{0})$ small enough and $R_{1}>R_{0}$ large enough such that if $u\in{\mathcal{N}}_{\epsilon,R}$ and $J_{\epsilon,R}(u)\leq c_{0}+\alpha_{0}$, then $Q_{\epsilon}(u)\in K_{\frac{\rho_{0}}{2}},\,\,\forall\epsilon\in(0,\epsilon_{1})$ and $R\geq R_{1}$.

Proof.

If the lemma is not true, then exist $\alpha_{n}\rightarrow 0,~{}\epsilon_{n}\to 0,R_{n}\to+\infty$ and $u_{n}\in{\mathcal{N}}_{\epsilon_{n},R_{n}}$ such that

[TABLE]

and

[TABLE]

Since $c_{\epsilon_{n},R_{n}}\geq c_{0}$, the above inequality gives

[TABLE]

and so,

[TABLE]

Setting the functional $\Psi_{\epsilon_{n},R_{n}}:H_{0}^{1}(B_{R_{n}})\to\mathbb{R}$ by

[TABLE]

we derive

[TABLE]

A simple computation ensures that there is $\beta>0$, which is independent of $n$, such that

[TABLE]

otherwise $c_{\epsilon_{n},R_{n}}\to 0$, which is absurd. With this information in our hands, we can apply the Ekeland Variational principal found in [13, Theorem 8.5] to assume, without loss of generality, that $\|J^{\prime}_{\epsilon_{n},B_{R_{n}}}(u_{n})\|\to 0$ as $n\to+\infty$.

By [1, Section 6], we need to consider the following two cases:

• (a)

$u_{n}\rightarrow u\neq 0$ in $L^{2}(\mathbb{R}^{N})$,

or

• (b)

There exists $\{y_{n}\}\subset\mathbb{R}^{N}$ such that $\vartheta_{n}=u_{n}(.+y_{n})\to\vartheta\not=0$ in $L^{2}(\mathbb{R}^{N})$.

Since $J_{\epsilon_{n},R_{n}}(u_{n})=\displaystyle\frac{1}{2}\int_{B_{R_{n}}(0)}|u_{n}|^{2}\geq c_{0}>0$, we derive that $\displaystyle\liminf_{n\rightarrow\infty}t_{n}>0$. Hence, the above conclusion ensures that

• ($a^{\prime}$)

$u_{n}\rightarrow u\neq 0$ in $L^{2}(\mathbb{R}^{N})$, for some $u\in H^{1}(\mathbb{R}^{N})$

or

• ($b^{\prime}$)

There exists $(y_{n})\subset\mathbb{R}^{N}$ such that $v_{n}=u_{n}(.+y_{n})\to v\not=0$ in $L^{2}(\mathbb{R}^{N})$, for some $v\in H^{1}(\mathbb{R}^{N})$.

If $(a^{\prime})$ holds, we have that

[TABLE]

From this, $Q_{\epsilon_{n}}(u_{n})\in K_{\frac{\rho_{0}}{2}}$ for $n$ large enough, which is a contradiction.

Now, if $(b^{\prime})$ holds, we must distinguish two cases:

• (I)

$|\epsilon_{n}y_{n}|\rightarrow+\infty$

or

• (II)

$\epsilon_{n}y_{n}\rightarrow y$ for some $y\in\mathbb{R}^{N}$, for some subsequence.

If $(I)$ holds, we have that $J^{\prime}_{\infty}(v)v\leq 0$. Thus, there is $s\in(0,1]$ such that $sv\in{\mathcal{N}}_{\infty}$

[TABLE]

which contradicts (2.3).

Now, if (II) holds, the previous argument yields

[TABLE]

where $c_{V(y)}$ is the mountain pass level of the functional $J_{V(y)}:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ given by

[TABLE]

One can see that

[TABLE]

where

[TABLE]

If $V(y)>1$, as in [1], it is possible to prove that $c_{V(y)}>c_{0}$, which contradicts (3.8). Then $V(y)=1$ and $y=z_{i}$ for some $i=1,...,l$. Hence

[TABLE]

from where it follows that $Q_{\epsilon_{n}}(u_{n})\in K_{\frac{\rho_{0}}{2}}$ for $n$ large, which is absurd, because we are assuming that $Q_{\epsilon_{n}}(u_{n})\not\in K_{\frac{\rho_{0}}{2}}$. This finishes the proof. ∎

Next, we specify the following symbols.

[TABLE]

Lemma 3.6**.**

Given $\gamma\in(0,(c_{\infty}-c_{0})/2)>0$, there exist $\epsilon_{2}\in(0,\epsilon_{1})$ small enough such that

[TABLE]

for all $\epsilon\in(0,\epsilon_{2})$ and $R\geq R_{1}=R_{1}(\epsilon)>R_{0}$.

Proof.

Let $u\in H^{1}(\mathbb{R}^{N})$ be a ground state solution for $J_{0}$, i.e.,

[TABLE]

Hereafter, for each $w\in D(J_{0})$, $J^{\prime}_{0}(w):H_{c}^{1}(\mathbb{R}^{N})\to\mathbb{R}$ means the functional given by

[TABLE]

and

[TABLE]

If $\|J^{\prime}_{0}(w)\|$ is finite, then $J_{0}^{\prime}(w)$ may be extended to a bounded operator in $H^{1}(\mathbb{R}^{N})$, and so, it can be seen as an element of $(H^{1}(\mathbb{R}^{N}))^{\prime}$.

For any $1\leq i\leq l$, there is $\epsilon_{1}>0$ such that

[TABLE]

Now, we fix $R>R_{1}=R_{1}(\epsilon)$ and $t_{\epsilon,R}>0$ such that function $u^{i}_{\epsilon,R}(x)=t_{\epsilon,R}\varphi_{R}(x)u(x-\frac{z_{i}}{\epsilon})\in{\mathcal{N}}_{\epsilon,R}$,

[TABLE]

and

[TABLE]

Here, $\varphi_{R}(x)=\varphi(\frac{x}{R})$ with $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$, $0\leq\varphi(x)\leq 1$ for all $x\in\mathbb{R}^{N}$, $\varphi(x)=1$ for $x\in B_{1/2}(0)$ and $\varphi(x)=0$ for $x\in B^{c}_{1}(0)$. Therefore,

[TABLE]

from it follows that

[TABLE]

Then, decreasing $\alpha_{0}$ if necessary,

[TABLE]

which is the first inequality. To obtain the second one, note that if $u\in\partial\Omega^{i}_{\epsilon,R}$, then

[TABLE]

that is, $Q_{\epsilon,R}(u)\not\in K_{\frac{\rho_{0}}{2}}$. Thus, from Lemma 3.5,

[TABLE]

and so

[TABLE]

Consequently, from (3.9)-(3.10),

[TABLE]

and the results are derived by fixing $\epsilon_{2}\in(0,\epsilon_{1})$. ∎

Theorem 3.1**.**

There are $\epsilon_{*}\in(0,\epsilon_{2})$ small enough and $R_{1}=R_{1}(\epsilon)>R_{0}$ large enough such that $J_{\epsilon,R}$ has at least $l$ nontrivial critical points for $\epsilon\in(0,\epsilon_{*})$ and $R\geq R_{1}$. Moreover, all of the solutions are positive.

Proof.

From Lemma 3.6, there exist $0<\epsilon_{*}<\epsilon_{2}$ small enough and $R_{1}>R_{0}$ large enough such that

[TABLE]

Arguing as in [2, Proof of Theorem 2.1], the above inequality permits to use the Ekeland’s variational principle to get a $(PS)_{\alpha^{i}_{\epsilon,R}}$ sequence $(u^{i}_{n})\subset\Omega^{i}_{\epsilon,R}$ for $J_{\epsilon,R}$. Noting that $\alpha^{i}_{\epsilon,R}<c_{0}+\gamma$, from Lemma 3.4 there exists $u^{i}$ such that $u_{n}^{i}\rightarrow u^{i}$ in $H_{0}^{1}(B_{R}(0))$. So

[TABLE]

Since

[TABLE]

We deduce that $u^{i}\neq u^{j}$ for $i\neq j$ for $1\leq i,j\leq l$. Hence $J_{\epsilon,R}$ possesses at least $l$ nontrivial critical points for all $\epsilon\in(0,\epsilon_{*})$ and $R\geq R_{1}$. Finally, decreasing $\gamma$ and increasing $R_{1}$ if necessary, we can assume that

[TABLE]

The above inequality permits to conclude that all of the solutions do not change sign, and as $f(t)=t\log t^{2}$ is an odd function, we can assume that they are nonnegative. Now, the positivity of the solutions in $B_{R}(0)$ follows by maximum principle. ∎

4 Existence of solution for original problem

In the following, for each $i\in\{1,...,\l\}$ and $\epsilon\in(0,\epsilon_{*})$, we set $R_{n}\to+\infty$ and $u_{n}^{i}=u_{\epsilon,R_{n}}^{i}$ be a solution obtained in Theorem 3.1. Then,

[TABLE]

and

[TABLE]

Proposition 4.1**.**

There exists $u^{i}\in H^{1}(\mathbb{R}^{N})$ such that $u^{i}_{n}\rightharpoonup u^{i}$ in $H^{1}(\mathbb{R}^{N})$ and $u^{i}\not=0$ for all $i\in\{1,..,l\}$.

Proof.

Since $(\alpha^{i}_{\epsilon,n})$ is a bounded sequence, it is easy to check that $(u^{i}_{n})$ is a bounded sequence. Hence, we may assume that $u^{i}_{n}\rightharpoonup u^{i}$ for some $u^{i}\in H^{1}(\mathbb{R}^{N})$. Arguing by contradiction, we assume that there is $i_{0}\in\{1,..l\}$ such that $u^{i_{0}}=0$. In the sequel $(u_{n})$ and $(\alpha_{n})$ denote $(u^{i}_{n})$ and $(\alpha^{i}_{\epsilon,R_{n}})$ respectively.

To proceed further we need to use the Concentration Compactness Principle, due to Lions [7], employed to the following sequence

[TABLE]

This principle assures that one and only one of the following statements holds for a subsequence of $(\rho_{n})$, still denoted by itself:

(Vanishing)

[TABLE]

(Compactness)

There exists a sequence of points $(y_{n})\subset\mathbb{R}^{N}$ such that for all $\eta>0$, there exists $K>0$ such that

[TABLE]

(Dichotomy)

There exist $(y_{n})\subset\mathbb{R}^{N}$, $\alpha\in(0,1)$, $K_{1}>0$, $K_{n}\to+\infty$ such that the functions $\displaystyle\rho_{1,n}(x):=\chi_{B_{K_{1}}(y_{n})}(x)\rho_{n}(x)$ and $\displaystyle\rho_{2,n}(x):=\chi_{B_{K_{n}}^{c}(y_{n})}(x)\rho_{n}(x)$ satisfy

[TABLE]

Our objective is to show that $(\rho_{n})$ verifies the Compactness condition and in order to do so we act by excluding the others two possibilities. But this fact will lead to a contradiction, showing the proposition.

The vanishing case (4.1) can not occur, otherwise we conclude that $|u_{n}|_{p}\rightarrow 0$, and so, $F^{\prime}_{2}(u_{n})u_{n}\to 0$ in $L^{1}(\mathbb{R}^{N})$. Arguing as in the previous section, it is possible to prove that $u_{n}\to 0$ in $H^{1}(\mathbb{R}^{N})$. However, this convergence contradicts the fact that $\alpha_{n}\geq C_{1}$ for all $n\in\mathbb{N}$, see Lemma 3.1.

Let us show that Dichotomy also does not hold. Suppose that this is not the case. Under this assumption, we claim that $(y_{n})$ is unbounded, because otherwise, in this case, using the fact that $|u_{n}|_{L^{2}(\mathbb{R}^{N})}\not\to 0$, the first convergence in (4.3) leads to

[TABLE]

for some $\delta>0$ and for sufficiently large $n$. Then, picking $R^{\prime}>0$ such that $B_{K_{1}}(y_{n})\subset B_{R^{\prime}}(0)$, for all $n\in\mathbb{N}$, it follows that

[TABLE]

Since $u_{n}\to 0$ in $L^{2}(B_{R^{\prime}}(0))$, the above inequality is impossible. Thereby $(y_{n})$ is an unbounded sequence. In what follows, we set

[TABLE]

Hence $(v_{n})\subset H^{1}(\mathbb{R}^{N})$ is bounded and, up to subsequence, we may assume that $v_{n}\rightharpoonup v$ and by the first part of (4.3) we have $v\not\equiv 0$.

Claim 4.1**.**

$F^{\prime}_{1}(v)v\in L^{1}(\mathbb{R}^{N})$* and $J_{\infty}^{\prime}(v)v\leq 0.$*

Note that, if $\varphi\in C^{\infty}_{0}(\mathbb{R}^{N})$, $0\leq\varphi\leq 1$, $\varphi\equiv 1$ in $B_{1}(0)$ and $\varphi\equiv 0$ in $B_{2}(0)^{c}$, defining $\varphi_{R}:=\varphi(\cdot/R)$ and $v=\varphi_{R}(\cdot-y_{n})u_{n}$, the following equality holds

[TABLE]

[TABLE]

Fixing $R$ and passing to the limit in the above equality when $n\rightarrow\infty$ we get

[TABLE]

Now, the claim follows, using that $F^{\prime}_{1}(t)t\geq 0$ for all $t\in\mathbb{R}$, and applying Fatou’s lemma in the last inequality, as $R\rightarrow+\infty$.

Therefore, there is $t_{\infty}\in(0,1]$ such that $t_{\infty}v\in{\mathcal{N}}_{\infty}$, and so,

[TABLE]

which contradicts the fact that $\gamma<c_{\infty}-c_{0}$. Thus, in any case, Dichotomy does not occur and, actually, Compactness must hold. To reach our goal let us state the last claim.

Claim 4.2**.**

The sequence of points $(y_{n})\subset\mathbb{R}^{N}$ in (4.2) is bounded.

The proof of this claim consists in assuming by contradiction that the sequence of points $(y_{n})$ is unbounded. Then, up to subsequence, $|y_{n}|\to+\infty$, and we proceed as in the case of Dichotomy, where $(y_{n})$ was unbounded, reaching that $c_{0}+\gamma\geq c_{\infty}$.

In view of Claim 4.2, for a given $\eta>0$, there exists $R>0$ such that, by (4.2),

[TABLE]

or equivalent to

[TABLE]

where $b=\displaystyle\sup_{n\in\mathbb{N}}|u_{n}|^{2}_{2}$. Then, for $R_{1}\geq\max\{R,R^{\prime}_{0}\}$, due to the convergence $u_{n}\to 0$ in $L^{2}(B_{R_{1}}(0))$, there exists $n_{0}\in\mathbb{N}$ such that

[TABLE]

Then, by (4.6) and (4.7), it follows that if $n\geq n_{0}$,

[TABLE]

for some $C$ that does not depend on $\eta$. As $\eta$ is arbitrary, we can conclude that $u_{n}\to 0$ in $L^{2}(\mathbb{R}^{N})$. Since $(u_{n})$ is bounded in $H^{1}(\mathbb{R}^{N})$, by interpolation on the Lebesgue spaces, it follows that

[TABLE]

However, this limit implies that $J_{\epsilon,R_{n}}(u_{n})=\alpha_{n}\to 0$, which is impossible, because $\alpha_{n}\geq c_{\epsilon}>0$ for all $n\in\mathbb{N}$. ∎

As an immediate consequence of Proposition 4.1, we have the corollary.

Corollary 4.1**.**

For each sequence $(u^{i}_{n})\subset H^{1}(\mathbb{R}^{N})$ given in Proposition 4.1 and for small $\epsilon\in(0,\epsilon_{*})$, we have that $u^{i}\not=0$ and $J^{\prime}_{\epsilon}(u^{i})v=0$ for all $v\in C_{0}^{\infty}(\mathbb{R}^{N})$. Moreover, the following limits hold

[TABLE]

Since

[TABLE]

we have that

[TABLE]

Proof.

By Proposition 4.1, we know that $u^{i}\not=0$ for all $i\in\{1,...,l\}$. The limit $u^{i}_{n}\to u^{i}$ in $L_{loc}^{q}(\mathbb{R}^{N})$ for all $q\in[2,2^{*})$ ensures that

[TABLE]

Since

[TABLE]

we can conclude that $J^{\prime}_{\epsilon}(u^{i})v=0$ for all $v\in C_{0}^{\infty}(\mathbb{R}^{N})$. Using the fact that $g(x)\to 0$ as $|x|\to+\infty$, it is easy to show that

[TABLE]

and

[TABLE]

The above limits ensure that (4.8) and (4.9) hold. ∎

4.1 Proof of Theorem 1.1

By Corollary 4.1, for each $i\in\{1,...,\l\}$ and $\epsilon\in(0,\epsilon_{*})$, there is a solution $u^{i}\in H^{1}(\mathbb{R}^{N})\backslash\{0\}$ for problem (1.3) such that

[TABLE]

Since

[TABLE]

it follows that $u^{i}\not=u^{j}$ for $i\not=j$. Due to a change of variable, the functions $v^{i}(x)=u^{i}(x/\epsilon),\forall x\in\mathbb{R}^{N}$, $i\in\{1,...,\l\}$ are $l$ positive solutions of problem (1.2).