Positive multi-peak solutions for a logarithmic Schrodinger equation

TL;DR

This paper proves the existence and local uniqueness of positive multi-peak solutions for a singularly perturbed logarithmic Schrödinger equation using Lyapunov-Schmidt reduction and a novel norm to handle the nonlinearity.

Contribution

It introduces a new analytical approach to study multi-peak solutions for the logarithmic Schrödinger equation, a problem not previously addressed with reduction techniques.

Findings

Existence of positive multi-peak solutions under certain conditions on V(x)

Development of a new norm to manage the logarithmic nonlinearity

Establishment of local uniqueness of solutions using Pohozaev identities

Abstract

In this manuscript, we consider the logarithmic Schr\"{o}dinger equation \begin{eqnarray*} -\varepsilon^2\Delta u+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where $N\geq3$, $\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schr\"{o}dinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.

Equations

$$-{\epsilon }^2\Delta u+V(x)u=u\log u^2,u>0,$$

$$i\epsilon {\partial }_{t}u+\frac{{\epsilon }^2}{2}△u-V(x)u+u\log u^2=0.$$

$${\epsilon }^2{\int }_{{\mathbb{R}}^N}\nabla u\nabla \psi +{\int }_{{\mathbb{R}}^N}V(x)u\psi ={\int }_{{\mathbb{R}}^N}u\psi \log u^2,\text{leavevmode}\text{mbox}forany\text{leavevmode}\psi \in H^1({\mathbb{R}}^N).$$

$$I_{\epsilon }(u)=\frac{{\epsilon }^2}{2}{\int }_{{\mathbb{R}}^N}|\nabla u{|}^2+\frac{1}{2}{\int }_{{\mathbb{R}}^N}(V(x)+1)u^2-\frac{1}{2}{\int }_{{\mathbb{R}}^N}{u}^2\log u^2,u\in H^1({\mathbb{R}}^N).$$

$$\int_{\mathbb{R}^{N}}u^{2}\log u^{2}\leq\frac{a^{2}}{\pi}\|\nabla u\|^{2}_{2}+\Big{(}\log\|u\|^{2}_{2}-N(1+\log a)\Big{)}\|u\|^{2}_{2},\ u\in H^{1}(\mathbb{R}^{N}),\ a>0,$$

$$\displaystyle u(x)=\left\{\begin{array}[]{ll}\big{(}\sqrt{x}\log x\big{)}^{-1},&x\geq 3;\\ 0,&x\leq 2.\end{array}\right.$$

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

$$\underset{|x|\to \infty }{\lim }V(x)=V_{\infty }\text{leavevmode}\text{mbox}and\text{leavevmode}0<V(x)<V_{\infty }\text{leavevmode}\text{mbox}forany\text{leavevmode}x\in {\mathbb{R}}^N.$$

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

$$\nabla V(\xi_{j})=0,\leavevmode\nobreak\ det\Big{(}\big{(}\frac{\partial^{2}V(\xi_{j})}{\partial\xi_{j,i}\partial\xi_{j,l}}\big{)}_{1\leq i,l\leq N}\Big{)}\neq 0,\leavevmode\nobreak\ \mbox{for any}\leavevmode\nobreak\ j=1,\cdots,k.$$

$$y_{\epsilon ,j}\to {\xi }_j,\text{leavevmode}\text{mbox}as\text{leavevmode}\epsilon \to 0.$$

$$|u_{\epsilon }(x)|\le \tau ,$$

$${\int }_{{\mathbb{R}}^N}({\epsilon }^2|\nabla u_{\epsilon }{|}^2+u_{\epsilon }^2)\le C{\epsilon }^N.$$

$$\langle u,v\rangle_{\varepsilon}=\int_{\mathbb{R}^{N}}\Big{(}\varepsilon^{2}\nabla u\cdot\nabla v+(V(x)+1)uv\Big{)},\leavevmode\nobreak\ H_{\varepsilon}=\big{\{}u\in H^{1}(\mathbb{R}^{N}):\|u\|_{\varepsilon}:=\langle u,u\rangle_{\varepsilon}^{1/2}<\infty\big{\}}.$$

$$u_{\epsilon }=\sum _{j=1}^k{U}_{\epsilon ,y_j}+\phi ,$$

$$\{\begin{array}{l}{L}_{\epsilon }\phi =l_{\epsilon }+R_{\epsilon }(\phi ),x\in {\mathbb{R}}^N,\\ \phi \in H^1({\mathbb{R}}^N),\end{array}$$

$$\Vert R_{\epsilon }(\phi ){\Vert }_{\epsilon }=o(\Vert \phi {\Vert }_{\epsilon }).$$

$$\{\begin{array}{l}{L}_{\epsilon }\phi =l_{\epsilon },x\in {\mathbb{R}}^N,\\ {\phi }_{\epsilon }\in H^1({\mathbb{R}}^N),\end{array}$$

$$\left\{\phi \in H^1({\mathbb{R}}^N):\Vert \phi {\Vert }_{\epsilon }\le {\epsilon }^{\tau }\Vert l_{\epsilon }{\Vert }_{\epsilon },0<\tau <1\right\}$$

$$|R_{\varepsilon}(\varphi)|=O\Big{(}\varphi^{2}\big{(}\sum^{k}_{j=1}U_{\varepsilon,y_{j}}\big{)}^{-1}\Big{)}.$$

$$\|\varphi\|_{*}=\sup_{x\in\mathbb{R}^{N}}\Big{(}\sum_{j=1}^{k}e^{-\frac{|x-y_{j}|^{2}}{2\varepsilon^{2}}}\Big{)}^{-1}|\varphi(x)|,$$

$${\wp }_{\epsilon }:=\left\{\phi \in H_{\epsilon }:\Vert \phi {\Vert }_{\ast }\le \frac{1}{|\ln \epsilon {|}^{1-\theta }}\right\},\text{leavevmode}\text{mbox}withsomesmall\text{leavevmode}\theta >0.$$

$$V({\xi }_1)=\underset{{\mathbb{R}}^N}{\inf }V(x)\text{leavevmode}\text{mbox}and\text{leavevmode}V(x)>V({\xi }_1)\text{leavevmode}\text{mbox}forany\text{leavevmode}x\in {\mathbb{R}}^N\setminus \{{\xi }_1\}.$$

$${\eta }_{\epsilon }=\frac{u_{\epsilon }^{(1)}-u_{\epsilon }^{(2)}}{\Vert u_{\epsilon }^{(1)}-u_{\epsilon }^{(2)}{\Vert }_{L^{\infty }({\mathbb{R}}^N)}}.$$

$$-\Delta u+wu=u\log u^2,u>0,\text{in}\text{leavevmode}{\mathbb{R}}^N.$$

$$\mathrm{Ker}\mathcal{L}=\mathrm{span}\left\{\frac{\partial U}{\partial x_j}:1\le j\le N\right\},$$

$$\mathcal{L}\phi \equiv -\Delta \phi +(\omega -2-2\log U)\phi ,\text{leavevmode}\text{mbox}for\text{leavevmode}\phi \in H^1({\mathbb{R}}^N).$$

$$U_{\epsilon ,y_j}(x)=e^{\frac{V(y_j)+N}{2}}{e}^{-\frac{|x-y_j{|}^2}{2{\epsilon }^2}},$$

$$-{\epsilon }^2\Delta U_{\epsilon ,y_j}(x)+V(y_j)U_{\epsilon ,y_j}(x)=U_{\epsilon ,y_j}(x)\log U_{\epsilon ,y_j}^2(x)\text{in }{\mathbb{R}}^N.$$

Full text

[TABLE]

where $N\geq 3$, $\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schrödinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.

1. Introduction and main results

In this paper, we consider the following logarithmic Schrödinger equations

[TABLE]

where $\varepsilon>0$ is a parameter, $N\geq 3$.

Eq. (1.1) is closely related to the time-dependent logarithmic Schrödinger equations

[TABLE]

Eq. (1.2) was proposed by Bialynicki-Birula and Mycielski [6] as a model of nonlinear wave mechanics. This NLS Eq. (1.2) has wide applications in quantum optics [7], nuclear physics [18], geophysical applications of magma transport [16], effective quantum and gravity, theory of superfluidity, Bose-Einstein condensation and open quantum systems(see [28, 29] and the references therein). For the existence, stability of standing waves and the Cauchy problem in a suitable functional framework about Eq. (1.2), we can refer to [3, 4, 11, 12, 13].

We call $u\in H^{1}(\mathbb{R}^{N})$ a (weak) solution to Eq. (1.1) if it holds that

[TABLE]

From a variational point of view, the search of nontrivial solutions to (1.1) can be formally associated with the study of critical points of the functional on $H^{1}(\mathbb{R}^{N})$ defined by

[TABLE]

By using the following standard logarithmic Sobolev inequality (see Theorem 8.14 in [21])

[TABLE]

it is easy to see that $\displaystyle\int_{\mathbb{R}^{N}}u^{2}\log u^{2}<+\infty$ for all $u\in H^{1}(\mathbb{R}^{N})$, but there exists $u\in H^{1}(\mathbb{R}^{N})$ such that $\displaystyle\int_{\mathbb{R}^{N}}u^{2}\log u^{2}=-\infty$. For example, if $N=1$, $u$ is a smooth function satisfying

[TABLE]

One can verify directly that $u\in H^{1}(\mathbb{R}^{N})$ and $\displaystyle\int_{\mathbb{R}^{N}}u^{2}\log u^{2}=-\infty$. Thus, in general, $I_{\varepsilon}(u)$ fails to be finite and $C^{1}$ smooth on $H^{1}(\mathbb{R}^{N})$.

Due to this loss of smoothness, the classical critical point theory cannot be applied for $I_{\varepsilon}$. In order to study existence of solutions to logarithmic Schrödinger equation, several approaches were used so far in the literature as far as we know. For problem (1.1) with $\varepsilon=1$, Cazenave [11] worked in a suitable Banach space $W$ endowed with a Luxemburg type norm in order to make the functional $I_{1}:W\rightarrow\mathbb{R}$ well defined and $C^{1}$ smooth. In recent years, non-smooth critical point theory was applied , such as Squassina and Szulkin [24, 25] studied the following logarithmic Schrödinger equation

[TABLE]

where $V(x)$ and $Q(x)$ are spatially periodic. They showed the existence of ground state and infinitely many possibly sign-changing solutions, which are geometrically distinct under $\mathbb{Z}^{N}$-action. See also [14, 15, 20] for more non-smooth variational framework to logarithmic Schrödinger equation. At the same time, by using penalization technique, Tanaka and Zhang [27] obtained infinitely many multi-bump geometrically distinct positive solutions of (1.3). We also refer to [17] for the approach of using penalization. Another interesting work concerning with Eq. (1.1) with $\varepsilon=1$ is [23], by using the constrained minimization method, which avoided using Luxemburg type norm, non-smooth critical point theory and penalization technique. Here Shuai [23] proved directly the minimizers of $I_{1}(u)$ on a Nehari set or a sign changing Nehari set are indeed solutions by direction derivative.

Recently, problem (1.1) was studied in [1] if $V(x)$ is a continuous function with a global minimum. By using variational method developed by Szulkin in [26] for functionals which are sum of a ${C}^{1}$ functional with a convex lower semi-continuous functional, Alves et al in [1] proved, for $\varepsilon>0$ small enough, the existence of positive solutions and concentration around of a minimum point of $V(x)$. Later, Alves and Ji in [2] studied the existence of multiple solutions for problem (1.1) under the following conditions on potential $V(x)$:

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

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(II). There exist $l$ points $z_{1},\cdots,\ z_{l}$ in $\mathbb{R}^{N}$ such that

[TABLE]

They proved that for $\varepsilon>0$ small enough, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions, specifically, Eq. (1.1) has at least $l$ positive solutions for $\varepsilon$ small enough.

From the above results, we summarize that all existing results on logarithmic Schrödin-ger equations are obtained by variational methods. In this paper, we intend to study logarithmic Schrödinger equation (1.1) by Lyapunov-Schmidt reduction.

More precisely, we suppose that $V(x)\in C^{1}:\mathbb{R}^{N}\to\mathbb{R}$ satisfies the following conditions:

$(V_{1}).$ $V(x)\in L^{\infty}(\mathbb{R}^{N})$ and $0<\inf_{\mathbb{R}^{N}}V(x)\leq\sup_{\mathbb{R}^{N}}V(x)<\infty$;

$(V_{2}).$ There exist $k$ points $\xi_{1},\cdots,\xi_{k}$ such that

[TABLE]

Here we also give the definition of $k$-peak solutions of Eq. (1.1) as usual.

Definition A. Let $k\in\mathbb{N}$ and $\xi_{j}\in\mathbb{R}^{N}$ with $j=1,\cdots,k$. We say that $u_{\varepsilon}\in H^{1}(\mathbb{R}^{N})$ is a $k$-peak solution of (1.1) concentrated at $\xi_{1},\cdots,\xi_{k}$ if**

(i) $u_{\varepsilon}$ has $k$ local maximum points $y_{\varepsilon,j}\in\mathbb{R}^{N}$, $j=1,\ldots,k$, satisfying

[TABLE]

(ii) For any given $\tau>0$, there exists $R\gg 1$, such that**

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(iii) There exists $C>0$ such that**

[TABLE]

Our first result concerning on the existence of $k$-peak solutions to (1.1) is as follows.

Theorem 1.1**.**

Assume that $N\geq 3$, $(V_{1})$ and $(V_{2})$ holds. Then, Eq. (1.1) has a $k$-peak solution concentrated at $\xi_{1},\cdots,\xi_{k}$ for $\varepsilon>0$ sufficiently small.

Now we outline the main ideas and difficulties in the proof of Theorem 1.1. The basic idea is to use the unique positive solution to the limiting equation of (1.1) as a building block to construct solutions for (1.1). We first reduce the problem to a finite dimensional one by Lyapunov-Schmidt reduction. Since the singularity of the nonlinear term $u\log u^{2}$, traditional reduction method (for example refer to [5]) can’t be used directly, we make a few modifications.

Here we introduce some notations. Denote

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And then we will construct $k$-peak solutions of Eq. (1.1) of the forms

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where $U_{\varepsilon,y_{j}}$ is the solution of limiting equation of (1.1) which will be defined later. So, Eq. (1.1) can be rewritten as the following equation about $\varphi$:

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where the linear operator $L_{\varepsilon}$, the terms $l_{\varepsilon}$ and $R_{\varepsilon}(\varphi)$ are be defined in Section 2 Later.

In the traditional calculations, under the general $H^{1}(\mathbb{R}^{N})$ norm, we find

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Then, for $\varphi$ small, (1.4) can be seen as a perturbation of the following problem

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Suppose that $L_{\varepsilon}$ is a bounded invertible map in some suitable space, then (1.6) has a solution $\varphi_{\varepsilon}=L_{\varepsilon}^{-1}l_{\varepsilon}$. So we can use the contraction mapping theorem in the following small ball

[TABLE]

to solve (1.4). While, for the logarithmic Schrödinger equations (1.1),

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In the general $H^{1}(\mathbb{R}^{N})$ space, $\|R_{\varepsilon}(\varphi)\|_{\varepsilon}$ isn’t a higher order small term of $\|\varphi\|_{\varepsilon}$, that is, (1.5) doesn’t hold. To overcome this difficulty, we define a new type of norm

[TABLE]

where $\varphi\in H_{\varepsilon}$, $y_{j}\in B_{\delta}(\xi_{j})$, and restrict $\varphi$ in the the following space

[TABLE]

Then we conduct the contraction mapping in a small ball $S$(see (3.30)) endowed with the norm $\|\cdot\|_{*}$.

After this reduction progress, we only need to solve a finite dimensional problem about $y_{j}$. Different from the general minimum or maximum progress, inspired by [22], we use the Pohozaev identity of (1.1) to ensure the existence of $y_{j}$. And this methods allow the peak points $y_{j}$ of $u_{\varepsilon}$ can be the non-degenerate critical points of $V(x)$, not just minimum points or maximum points of $V(x)$.

We also consider the local uniqueness of the $k$-peak solution of (1.1).

Theorem 1.2**.**

Assume that $(V_{1})$ and $(V_{2})$ hold. If $u^{(i)}_{\varepsilon}$ with $i=1,2$ are the positive solution of (1.1) concentrated at $\xi_{1},\cdots,\xi_{k}$. Then $u_{\varepsilon}^{(1)}\equiv u_{\varepsilon}^{(2)}$ for $\varepsilon$ sufficiently small.

Remark 1.3**.**

In Theorem 1.2 with $k=1$, we find the uniqueness result about single-peak solution concentrated at a non-degenerate critical point of $V(x)$. On the other hand, the ground state of (1.1) must concentrate at a minimum point of $V(x)$. So if we impose an other condition on $V(x)$ as follows:

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Then the ground state of (1.1) is unique by Theorem 1.2.

We will prove Theorem 1.2 inspired by [8]. Let $u_{\varepsilon}^{(l)}$ with $l=1,2$ be two different positive solutions concentrated at $k$ points $\xi_{1},\cdots,\xi_{k}$. Set

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Then we prove $\|\eta_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}=o(1)$ to obtain a contradiction with $\|\eta_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}=1$. We will use the blow-up analysis and local Pohozaev type of identities to deal with the estimate near the concentrated points. But we will use the maximum principle for the calculations away from the concentrated points.

In this paper, we write $\int u$ to denote Lebesgue integrals over $\mathbb{R}^{N}$, unless otherwise stated, \|u\|_{p}=\big{(}\int u^{p}\big{)}^{\frac{1}{p}} and $\langle u,v\rangle=\int uv$. We will use $C$ to denote various positive constants, and $O(t)$, $o(t)$ and $o(1)$ to mean $|O(t)|\leq C|t|$, $o(t)/t\to 0$ as $t\to 0$ and $o(1)\rightarrow 0$ as $\varepsilon\rightarrow 0$, respectively.

The paper is organized as follows. In Section 2 we give some notations and preliminary estimates. In Section 3, we carry out the reduction argument. In Sections 4 and 5, we will complete the proofs of Theorems 1.1 and 1.2 correspondingly.

2. Preliminaries

From [14], we know that $U(x):=e^{\frac{w+N-|x|^{2}}{2}}$ is the unique positive solution of the following problem

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Furthermore, it is non-degenerate in $H^{1}(\mathbb{R}^{N})$ in the sense that

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where the linearized operator $\mathcal{L}:H^{1}(\mathbb{R}^{N})\to H^{1}(\mathbb{R}^{N})$ is defined as

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For any $y_{j}\in\mathbb{R}^{N}$ with $j=1,\cdots,k$, we denote

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which is the solution of

[TABLE]

The linearized operator of (2.1) at $U_{\varepsilon,y_{j}}(x)$ is $\mathcal{L}_{\varepsilon}:\equiv-\varepsilon^{2}\Delta+V(y_{j})-2(\log U_{\varepsilon,y_{j}}+1)$, whose kernel is

[TABLE]

We note $y=(y_{1},\cdots,y_{k})$ and

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Let $\xi_{j}(j=1,\cdots,k)$ be the critical points of $V(x)$, we want to construct a solution $u_{\varepsilon}$ to Eq. (1.1) of the form

[TABLE]

where $y_{\varepsilon,j}\in\mathbb{R}^{N},\ \varphi_{\varepsilon}\in E_{\varepsilon,y}$ satisfies

[TABLE]

Then $\varphi_{\varepsilon}$ satisfies the following equation:

[TABLE]

where

[TABLE]

[TABLE]

and

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The procedure to construct a $k$-peak solution for (1.1) consists of two steps:

Step (1). Finite dimensional reduction: We solve (2.2) up to an approximate kernel $K_{\varepsilon}$ of $L_{\varepsilon}$. That is, for any given $y_{j}\in\mathbb{R}^{N}(j=1,\cdots,k)$, we prove the existence of $\varphi_{\varepsilon}\in E_{\varepsilon,y}$, such that**

[TABLE]

Step (2). * Solve the finite dimensional problem. We need to choose $y_{j}$ suitably, such that all the constants $a_{\varepsilon,i,j}$ in (2.6) are zero.*

In order to use the contraction mapping theorem to carry out the reduction for $(\ref{new form equation})$, we need the following invertible result and estimate $\|l_{\varepsilon}\|_{\varepsilon}$ and $\|R_{\varepsilon}(\varphi_{\varepsilon})\|_{\varepsilon}$.

Proposition 2.1**.**

There exist $\varepsilon_{1},\delta_{1},\rho>0$, independent of $y_{j},\ j=1,\cdots,k$, such that for any $\varepsilon\in(0,\varepsilon_{1}]$, $\delta\in(0,\delta_{1})$ and $y_{j}\in B_{\delta}(\xi_{j})$, $P_{\varepsilon}L_{\varepsilon}$ is bijective in $E_{\varepsilon,y}$. Moreover, it holds

[TABLE]

with the projection $P_{\varepsilon}$ from $H^{1}(\mathbb{R}^{N})$ to $E_{\varepsilon,y}$ as follows:

[TABLE]

Proof.

We use a contradiction argument. Assume, on the contrary, that there exist $\varepsilon_{n}\to 0$, $\delta_{n}\to 0$, $y_{n,j}\in B_{\delta_{n}}(\xi_{j})$ and $\varphi_{n}\in E_{n}\equiv E_{\varepsilon_{n},y_{n,j}}$ such that

[TABLE]

Since the equality is homogeneous, we may assume, with no loss of generality, that $\|\varphi_{n}\|_{\varepsilon_{n}}=\varepsilon_{n}^{N/2}$. Using (2.8), we get

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On the other hand, for $R>0$ large enough, we have

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So,

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Combining with (2.9), we get

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To deduce contradiction from (2.10), we only need to prove

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For this purpose, we will discuss the local behaviors of $\varphi_{n}$ near each $y_{n,j}(j=1,\cdots,k)$. So we introduce

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Then, since $V(x)$ is bounded and $\inf_{\mathbb{R}^{N}}V>0$, we have

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Hence, up to a subsequence, we may assume that

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for some $\varphi_{j}\in H^{1}(\mathbb{R}^{N})$, we will prove $\varphi_{j}\equiv 0$. Define

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Now, for any $\widetilde{\phi}_{n,j}\in\widetilde{E}_{\varepsilon_{n}}$, by (2.8), it holds

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where $\phi_{n,j}(x)=\widetilde{\phi}_{n,j}(\frac{x-y_{n,j}}{\varepsilon_{n}})\in E_{\varepsilon_{n}}$. For any $\phi\in H^{1}(\mathbb{R}^{N})$, there exists $c_{\varepsilon_{n},i,j}\in\mathbb{R}$ satisfying

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If $\phi$ satisfies

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for $i=1,\cdots,N,\ j=1,\cdots,k$, then $c_{\varepsilon_{n},i,j}=0$. Inserting $\widetilde{\phi}_{n,j}$ into (2.12) and letting $n\rightarrow\infty$, we find

[TABLE]

where $U^{j}=U_{\varepsilon_{n},y_{n,j}}(\varepsilon_{n}y+y_{n,j})=e^{\frac{V(\xi_{j})+N-|x|^{2}}{2}}$ satisfies $-\triangle U^{j}+V(\xi_{j})U^{j}=U^{j}\log(U^{j})^{2}.$

Furthermore, we know

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And then (2.13) also holds for $\phi=\displaystyle\sum_{i=1}^{N}\frac{\partial U^{j}}{\partial x_{i}}$. Thus, (2.13) holds for any $\phi\in H^{1}(\mathbb{R}^{N})$. So we have

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Thus, the non-degeneracy of $U^{j}$ gives $\varphi_{j}=\displaystyle\sum_{i=1}^{N}c_{i}\frac{\partial U^{j}}{\partial x_{i}}$.

On the other hand, $\widetilde{\varphi}_{n,j}\in\widetilde{E}_{\varepsilon_{n}}$ implies $\langle\varphi_{j},\frac{\partial U^{j}}{\partial x_{i}}\rangle_{\varepsilon}=0$ for any $i=1,\cdots,N$. As a result, $\varphi_{j}=0$ and thus (2.11) follows. We complete the proof. ∎

Lemma 2.2**.**

Assume that $V$ satisfies (V1) and (V2). Then, there exists a constant $C>0$, independent of $\varepsilon,\delta$, such that for any $y_{j}\in B_{\delta}(\xi_{j})$ there holds

[TABLE]

Proof.

From $(\ref{zero order})$, for any $\eta\in H_{\varepsilon}$, we have

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As

[TABLE]

and

[TABLE]

then we get (2.14) from (2.15) and (2.16). ∎

Lemma 2.3**.**

It holds

[TABLE]

where $\wp_{\varepsilon}$ was defined in (1.9).

Proof.

First, by (2.5) and Taylor’s expansion, we find (1.7). Then we can obtain

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Thus we complete the proof. ∎

3. Finite dimensional reduction

In this section, we carry out the reduction argument. For any fixed $y_{j}\in B_{\delta}(\xi_{j}),\ j=1,\cdots,k$, we consider the following problem:

[TABLE]

Lemma 3.1**.**

It holds

[TABLE]

Proof.

Recall (2.4), since $V(x)\in C^{1}$ satisfies $(V_{1})$, we have

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Similar to (2.16),

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Thus we get \|l_{\varepsilon}\|_{*}=O\big{(}1\big{)}. Also by (2.5),

[TABLE]

Then we obtain \|R_{\varepsilon}(\varphi)\|_{*}=O\big{(}\|\varphi\|_{*}^{2}\big{)}. ∎

Proposition 3.2**.**

Assume $N\geq 3$, $u$ solves

[TABLE]

with $\varphi\in E_{\varepsilon,y}$ satisfying

[TABLE]

where $\theta>0$ is a small positive constant. Then it holds

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Proof.

From (2.7) and (3.3), we have

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Combining with the definition of $L_{\varepsilon}$ in (2.3), we get

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Then we note

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with

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Now we estimate each term of (3.4). We first give an elementary inequality

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which will be useful during the following process.

For $x\in\displaystyle\bigcap_{j=1}^{k}B^{c}_{R\varepsilon}(y_{j})$, we have

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Also $-\Delta\frac{1}{|z-x|^{N-2}}=\delta_{z}(x)$ in $\mathbb{R}^{N}$. Let $w(x)=\frac{\varepsilon^{4}}{|x-y_{j}|^{2}}e^{-\frac{|x-y_{j}|^{2}}{2\varepsilon^{2}}}$, then $-\Delta w(x)\geq Ce^{-\frac{|x-y_{j}|^{2}}{2\varepsilon^{2}}}$,

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This gives

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So we get

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For $x\in B_{R\varepsilon}(y_{j}),\ j=1,\cdots,k$, we have

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And then

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since $|z-y_{j}|\geq|z-x|-|x-y_{j}|\geq R\varepsilon$ for $z\in B^{c}_{2R\varepsilon}(x)$ and $x\in B_{R\varepsilon}(y_{j})$. We find

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where $\gamma>0$ small, $p=\frac{N-\gamma}{N-2}$ and $q=\frac{N-\gamma}{2-\gamma}$. So, by (3.7)-(3.9), we know

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We conclude from (3.6) and (3.10) that

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Now we estimate $u_{2}(x)$. By a fact that for any $\alpha>0$,

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which gives

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For $x\in\displaystyle\bigcap_{j=1}^{k}B^{c}_{R\varepsilon}(y_{j})$, by (3.5), we have

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while, if we denote $\frac{(1+\alpha)}{N(N-2)\omega_{N}}=C_{N,\alpha}$,

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Take $R^{2}>2N$, for $z\in B^{c}_{R\varepsilon}(y_{j})$, we get

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so we have

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Then, we find

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On the other hand, by (3.5),

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As a result,

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Combing (3.12) and (3.13), we get

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For $x\in B_{R\varepsilon}(y_{j}),\ j=1,\cdots,k$, similar to (3.7)-(3.9), we can get

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By (3.14) and (3.15), we finally get

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for suitably large $R$ and $\varepsilon$ small.

Next we estimate $u_{3}$. Recall (2.4), we denote

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and

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For $x\in\displaystyle\bigcap_{j=1}^{k}B^{c}_{\varepsilon\sqrt{|\ln\varepsilon|}}(y_{j})$, we find

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Then by (3.5), we get

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Similarly, as

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we also have

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These give us that

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Next, we consider the case $x\in B_{\varepsilon\sqrt{|\ln\varepsilon|}}(y_{j}),\ j=1,\cdots,k$.

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Also by (3.2), we have

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and

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where $\gamma>0$ small, $p=\frac{N-\gamma}{N-2}$ and $q=\frac{N-\gamma}{2-\gamma}$. So, we find

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From (3.17) and (3.18), we get

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By using (3.2) and a similar estimate to $u_{3}$, we can get

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Now similar to the estimate of $u_{1}$, we estimate $u_{5}$. First, by $(\ref{estimates of linear part})$, we know

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On the other hand, we have

[TABLE]

For $x\in\displaystyle\bigcap_{j=1}^{k}B^{c}_{R\varepsilon}(y_{j})$, let $w_{1}(x)=\frac{\varepsilon^{4}}{|x-y_{j}|}e^{-\frac{|x-y_{j}|^{2}}{2\varepsilon^{2}}}$, we find $-\Delta w_{1}(x)\geq C|x-y_{j}|e^{-\frac{|x-y_{j}|^{2}}{2\varepsilon^{2}}}$ and then

[TABLE]

This gives

[TABLE]

For $x\in\displaystyle\bigcup_{j=1}^{k}B_{R\varepsilon}(y_{j})$, we have

[TABLE]

and then

[TABLE]

Also, we find

[TABLE]

Then by (3.23)-(3.25), we know

[TABLE]

So from (3.22) and (3.26), we find

[TABLE]

Above all, from (3.4), (3.11), (3.16), (3.19), (3.20), (3.27), we get

[TABLE]

∎

Proposition 3.3**.**

Assume $N\geq 3$. Let $\delta>0$ be small such that $B_{\delta}(\xi_{i})\cap B_{\delta}(\xi_{j})=\emptyset$ for $i,j=1,\cdots,k$, $i\neq j$, there exists $\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0}]$, $y_{j}\in B_{\delta}(\xi_{j})$, there is a unique map $\varphi_{\varepsilon,y}:B_{\delta}(\xi_{j})\to H_{\varepsilon}$ with $y\mapsto\varphi_{\varepsilon,y}\in E_{\varepsilon,y}$ satisfying (3.1), where $y=(y_{1},\cdots,y_{k})$. Moreover,

[TABLE]

and

[TABLE]

Proof.

By Proposition 2.1, we can rewrite (3.1) as

[TABLE]

It follows from Proposition 2.1 and (2.14) that

[TABLE]

Now we will apply the contraction mapping theorem in the set

[TABLE]

endowed with the norm $\|\cdot\|_{*}$, where $\tau,\theta>0$ are some fixed small constants.

Then for any $\varphi_{1},\ \varphi_{2}\in S$, it holds

[TABLE]

where $\theta\in[0,1]$. For any $\varphi\in E_{\varepsilon,y}$, by Lemma 2.2 and Lemma 2.3, we get

[TABLE]

On the other hand, applying Lemma 3.2 to $u=B\varphi$, we have

[TABLE]

So we get $B\varphi\in S$. Then by the contraction mapping theorem, we conclude that for $\varepsilon,\ \delta$ sufficiently small, there exists $\varphi_{\varepsilon}\in E_{\varepsilon,y}$ depending on $y$ and $\varepsilon$, satisfying $\varphi_{\varepsilon}=B\varphi_{\varepsilon}$. Moreover, we know

[TABLE]

which gives

[TABLE]

∎

4. Proof of Theorem 1.1

Theorem 1.1 can be deduced from the following result.

Theorem 4.1**.**

Assume that $(V_{1})$ and $(V_{2})$ holds, $N\geq 3$. Then, for $\varepsilon>0$ sufficiently small, equation (1.1) has a solution of the form

[TABLE]

for some $y_{\varepsilon,j}\in B_{\delta}(\xi_{j})$, $\|\varphi_{\varepsilon}\|_{\varepsilon}=O(\varepsilon^{\frac{N}{2}+1})$ and $\|\varphi_{\varepsilon}\|_{*}\leq\frac{1}{|\ln\varepsilon|^{1-\theta}}$ with some small $\theta>0$.

First, Proposition 3.3 implies the existence of $\varphi_{\varepsilon}\in E_{\varepsilon,y}$ , such that

[TABLE]

for some constants $a_{\varepsilon,i,j}$. So we need to choose $y_{j}$ suitably such that $a_{\varepsilon,i,j}=0,\ i=1,\cdots,N,\ j=1,\cdots,k$.

The function in the right hand side of (4.1) belongs to

[TABLE]

Therefore, we want to prove the left hand side of (4.1) belongs to $E_{\varepsilon,y}$ , then the function in the right hand side of (4.1) must be zero.

We first use the notation that

[TABLE]

Then, for any $\eta\in H_{\varepsilon}$,

[TABLE]

Lemma 4.2**.**

Suppose that $y_{\varepsilon,j}$ with $j=1,\cdots,k$ satisfies

[TABLE]

Then $a_{\varepsilon,i,j}=0,\ i=1,\cdots,N,\ \ j=1,\cdots,k$.

Proof.

If (4.2) holds, then

[TABLE]

Which implies that $a_{\varepsilon,i,j}=0,\ i=1,\cdots,N,\ \ j=1,\cdots,k$. ∎

Proof of Theorem 4.1.

We only need to solve the algebraic equations (4.2). The main task is to find the main term for the function in the left hand side of (4.2). The procedure is that we first estimate the left hand side of (4.2) with $\varphi_{\varepsilon}=0$ . Then we show that the contribution of the error term $\varphi_{\varepsilon}$ to the function in the left hand side of (4.2) is negligible.

Denote $G_{\varepsilon,y}:=\displaystyle\sum_{j=1}^{k}U_{\varepsilon,y_{j}}$. From (2.1) and the symmetry of $U_{\varepsilon,y_{j}}$ we get

[TABLE]

and

[TABLE]

for some $c>0$. Moreover, similar to (2.16), we have

[TABLE]

From above, we obtain

[TABLE]

Now we show that the contribution of the error term $\varphi_{\varepsilon}$ to the function in the left hand side of (4.2) is negligible.

As $\varphi_{\varepsilon}\in E_{\varepsilon,y}$, for $i=1,\cdots,N,\ j=1,\cdots,k$, we have

[TABLE]

On the other hand,

[TABLE]

First from (2.1) and $\varphi_{\varepsilon}\in E_{\varepsilon,y}$, we have

[TABLE]

So, similar to (2.16), we get

[TABLE]

For the other term, we have

[TABLE]

So we get

[TABLE]

As a result, (4.2) is equivalent to

[TABLE]

By (4.3) and the assumption $(V_{2})$, we have

[TABLE]

Then (4.3) has a solution $y_{\varepsilon,j}\in B_{\delta}(\xi_{j})$. We complete the proof. ∎

5. Local uniqueness results

In this section, we prove the local uniqueness result Theorem 1.2. First, we give an important estimate on $|y_{\varepsilon,j}-\xi_{j}|$, which can be improved by using a class of Pohozaev type identities. And the crucial Pohozaev type identities we will use are as follows:

Proposition 5.1**.**

Let $u$ be a positive solution of Eq. (1.1). Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$. Then, for each $i=1,\cdots,N$, there hold

[TABLE]

where $\nu=(\nu_{1},\cdots,\nu_{N})$ is the unit outward normal of $\partial\Omega$.

Proposition 5.1 can be directly proved by multiplying both sides of Eq. (1.1) by $\frac{\partial u}{\partial x_{i}}$ and then integrating by parts. Next, similar to Proposition 2.2 in [19], we find

Lemma 5.2**.**

If $\varphi_{\varepsilon}$ in Theorem 4.1 satisfies $\|\varphi_{\varepsilon}\|_{\varepsilon}=o(\varepsilon^{\frac{N}{2}})$, then there exists a small constant $\tau>0$, such that

[TABLE]

Proposition 5.3**.**

Let $u_{\varepsilon}=\displaystyle\sum_{j=1}^{k}U_{\varepsilon,y_{\varepsilon,j}}+\varphi_{\varepsilon}$ be a solution of (1.1). Then

[TABLE]

Proof.

Let $u=u_{\varepsilon},\ \Omega=B_{\delta}(y_{\varepsilon,j})$ in (5.1), we obtain

[TABLE]

From Lemma 5.2, we have

[TABLE]

here and in what follows $\gamma>0$ denote a constant which may change from line to line. By (5.4), for $x\in\partial B_{\delta}(y_{\varepsilon,j})$, we find

[TABLE]

So, (5.3) equivalent to

[TABLE]

On the other hand,

[TABLE]

Here we use Lemma 5.2. Now, by the symmetry of $U_{\varepsilon,y_{\varepsilon,j}}$, we have

[TABLE]

By Hölder inequality and (3.28), we can get

[TABLE]

Inserting above into (5.6) and combine with (5.5), we obtain

[TABLE]

Then, for $l=1,\cdots,N$,

[TABLE]

So, combining the condition $(V_{2})$ and $\displaystyle\int_{B_{\delta}(y_{\varepsilon,j})}u_{\varepsilon}^{2}=O(\varepsilon^{N})$, we get (5.2). ∎

Lemma 5.4**.**

Assume $u_{\varepsilon}=\displaystyle\sum_{j=1}^{k}U_{\varepsilon,y_{\varepsilon,j}}+\varphi_{\varepsilon}$ be a solution of (1.1). Then

[TABLE]

Proof.

First, we know the following property

[TABLE]

As the proof of (5.7) is standard (see e.g. [9]), we omit the details. We mainly estimate $\langle L_{\varepsilon}\varphi_{\varepsilon},\varphi_{\varepsilon}\rangle$. From (2.2), we have

[TABLE]

where $L_{\varepsilon}\varphi_{\varepsilon},\ l_{\varepsilon}$ and $R_{\varepsilon}(\varphi_{\varepsilon})$ are defined in (2.3)–(2.5). By (2.16), we get

[TABLE]

Under the condition $(V_{2})$, we obtaian

[TABLE]

So, we find

[TABLE]

By (2.5) and (3.29), we have

[TABLE]

Combining (5.2), (5.7)-(5.9), we get

[TABLE]

∎

Now we devoted to prove Theorem 1.2. We argue by way of contradiction. Assume $u_{\varepsilon}^{(i)}=\displaystyle\sum_{j=1}^{k}U_{\varepsilon,y^{(i)}_{\varepsilon,j}}+\varphi_{\varepsilon}^{(i)}(i=1,2)$ are two distinct solutions concentrating around $\xi_{j}$. Set

[TABLE]

then

[TABLE]

where

[TABLE]

It is clear that $\|\eta_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}=1$. We will prove that

[TABLE]

to obtain a contradiction. For fixed $j\in\{1,\cdots,k\}$, set

[TABLE]

To prove (5.11), we will prove that $\|\eta_{\varepsilon,j}\|_{L^{\infty}(B_{R}(0))}=o(1)$ and $\|\eta_{\varepsilon,j}\|_{L^{\infty}(\mathbb{R}^{N}\backslash B_{R}(0))}=o(1)$ holds separately.

First we study the asymptotic behavior of $\eta_{\varepsilon,j}$.

Proposition 5.5**.**

There exist $d_{\beta,j}\in\mathbb{R}$, $\beta=1,\cdots,N,\ j=1,\cdots,k$, such that (up to a subsequence)

[TABLE]

as $\varepsilon\to 0$, where $U^{j}$ solves

[TABLE]

.

Proof.

We will prove that the limiting function of $\eta_{\varepsilon,j}$ belongs to the kernel of the linear operator associated to $U^{j}$.

In view of $\|\eta_{\varepsilon,j}\|_{L^{\infty}(\mathbb{R}^{N})}\leq 1$, the elliptic regularity theory implies that $\eta_{\varepsilon,j}\in C_{\text{\rm loc}}^{1,\theta}(\mathbb{R}^{N})$ with respect to $\varepsilon$ for some $\theta\in(0,1)$. As a consequence, we assume (up to a subsequence) that

[TABLE]

We claim that $\eta_{j}$ satisfies

[TABLE]

Then by the fact that that $U^{j}$ is nondegenerate, we have $\eta_{j}=\sum_{\beta=1}^{N}d_{\beta,j}\frac{\partial U^{j}}{\partial x_{\beta}}$ for some $d_{\beta,j}\in\mathbb{R}$ ($\beta=1,\cdots,N$), and thus Proposition 5.5 is proved.

Next, we prove (5.12). From (5.10), we have $\eta_{\varepsilon,j}$ satisfies

[TABLE]

Now we estimate $C_{\varepsilon}(\varepsilon x+y^{(1)}_{\varepsilon,j})$. From (5.2),

[TABLE]

where $0<\theta<1,\ t=1,\cdots,k$ and $U_{y_{t}}$ satisfies

[TABLE]

For simplicity, here and what follows, we denote

[TABLE]

Then,

[TABLE]

So, for $x\in B_{d}(y^{(1)}_{\varepsilon,j})$

[TABLE]

Then, we know

[TABLE]

where

[TABLE]

and $\gamma>0$ is a constant. Now recall (5.13), we know

[TABLE]

Letting $\varepsilon\rightarrow 0$ in (5.16), we obtain (5.12). The proof is completed. ∎

Next, similar to Lemma 5.2, we find

Lemma 5.6**.**

There exists a small constant $d>0$, such that

[TABLE]

Proposition 5.7**.**

Let $d_{\beta,j}$ be defined as in Proposition 5.5. Then

[TABLE]

Proof.

Applying (5.1) to $u_{\varepsilon}^{(1)}$ and $u_{\varepsilon}^{(2)}$ with $\Omega=B_{d}(y^{(1)}_{\varepsilon,j})$, where $d$ is chosen such that $0<d<\min_{i\neq j}|y^{(1)}_{\varepsilon,i}-y^{(1)}_{\varepsilon,j}|$, we have

[TABLE]

where $1\leq i\leq N$ and

[TABLE]

with $\tilde{u}_{\varepsilon}=u_{\varepsilon}^{(1)}+\theta(u_{\varepsilon}^{(1)}-u_{\varepsilon}^{(2)}).$ By (5.15), we have for $x\in\partial B_{d}(y^{(1)}_{\varepsilon,j})$,

[TABLE]

Notice that $|\eta_{\varepsilon}|\leq 1$, so

[TABLE]

By (5.4) and Lemma 5.6, we have

[TABLE]

and

[TABLE]

So, (5.17) equivalent to

[TABLE]

As $V(x)$ satisfies $(V_{2})$, for $l=1,\cdots,N$, we have

[TABLE]

From (5.15), we have

[TABLE]

Then, we find

[TABLE]

By (5.2) and Proposition 5.5, we have

[TABLE]

Similarly, as $\|\eta_{\varepsilon}\|=O(\varepsilon^{\frac{N}{2}})$ and $\|\varphi_{\varepsilon}\|=O(\varepsilon^{\frac{N}{2}+2})$, we have

[TABLE]

and

[TABLE]

Combining above with (5.18), (5.19), we have

[TABLE]

Then $d_{l,j}=0$ for $l=1,\cdots,N,\ j=1,\cdots,k,$ since $U^{j}$ is a radially symmetric decreasing function. ∎

Proof of Theorem 1.2.

Propositions 5.5 and 5.7 show that

[TABLE]

for any $j=1,\cdots,k$, which means

[TABLE]

On the other hand, by using maximum principle, we can prove

[TABLE]

we can refer to [8, Proposition 3.5] for the similar detail proof. Consequently, we get (5.11), which contradict to $\|\eta_{\varepsilon}\|_{\infty}=1.$ The proof of local uniqueness is completed. ∎

Acknowledgments

The authors would like to thank Profs. Shuangjie Peng and Shusen Yan for many useful suggestions during the preparation of this paper. The first author (Peng Luo) was partially supported by NSFC grants (No.11701204, No.11831009)