Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

Abstract

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For $s=1$, there many articles have analyzed the limit behavior of minimizers when $\eta >0$ as $b\to 0^{+}$ or $b>0$ as $\eta \to 0^{+}$. When the equation involves a varying non-local term ${\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^s$, we give a detailed limit behavior analysis of constrained minimizers for any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$. The present paper obtains an interesting result on this topic and enriches the conclusions of previous works.

1. Introduction and Main Results

We consider the following Kirchhoff-type equation with a varying non-local term

$$-\left(\eta +b{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^s\right)\Delta u+V\left(x\right)u=\mu u+\lambda {\left|u\right|}^{p}u,$$

(1)

where $b>0$ is a constant, parameters $\eta \ge 0,\lambda >0$, exponents $s>0$, $0<p<4$ and $\mu$ is a Lagrange multiplier. The $b{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^s$ in (1) arises as a varying non-local term.

In recent years, there have been many articles involved in different types of varying non-local problems similar to (1) such as the model

$$\left\{\begin{array}{cc}-C{\left({\int }_{\Omega }{|\nabla u|}^{2}dx\right)}^s\Delta u=h(x,u){\left({\int }_{\Omega }f(x,u)dx\right)}^r,\hfill & \hfill x\in \Omega ,\\ u=0,\hfill & \hfill x\in \partial \Omega ,\end{array}\right.$$

which mainly studied the existence of solutions by using variational theory and analytical methods, as seen in [1,2,3,4].

Especially for $s=1$ in (1), the Kirchhoff-type constrained minimization problems are related to

$$-\left(\eta +b{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)\Delta u+V\left(x\right)u=\mu u+\lambda {\left|u\right|}^{p}u$$

which have attracted a significant number of mathematicians to study their existence, non-existence, uniqueness and limit behavior of constraint minimizers. More detailed, for $V\left(x\right)=0$, Ye [5,6] obtained some results of existence and nonexistence on constraint minimizers. Zeng and Zhang [7] proved the local uniqueness of minimizer, and then they [8] provided an analysis of asymptotic behavior for minimizers when $V\left(x\right)$ satisfies periodic potential. Guo, Zhang and Zhou [9] analyzed the existence and limit behavior of minimizers if the trapping potential $V\left(x\right)>0$ satisfies ${lim\; inf}_{\left|x\right|\to +\infty }V\left(x\right)=\infty$. In papers [10,11,12,13], the authors studied the existence and non-existence of constraint minimizers for the Kirchhoff-type energy functional with a $L^2$-subcritical term. Also for $V\left(x\right)$ being a polynomial function, the articles [14,15] obtained the limit behavior of $L^2$-norm solutions when $\eta >0$ as $b\to 0^{+}$ or $b>0$ as $\eta \to 0^{+}$.

Coincidentally for $s=0$ and ${\mathbb{R}}^3$ replaced by ${\mathbb{R}}^2$, the (1) comes from an interesting physical context, which is associated with the well known Bose–Einstein condensates (BECs). The mathematical theory study of BECs can be described by a Gross–Pitaevskii (GP) functional [16,17], which is related to the elliptic equation

$$\Delta u+V\left(x\right)u=\mu u+{\lambda \left|u\right|}^{p}u.$$

There are many researchers devoted to exploring the properties of the ground states for the GP functional related to the above elliptic equation. More precisely, when the external trapping potentials $V\left(x\right)$ are in the forms of polynomial, ring-shaped, multi-well, periodic and sinusoidal, the articles [18,19,20,21,22] gave the existence, non-existence and mass concentration behavior analysis of the ground states. If $V\left(x\right)$ behaves like logarithmic or homogeneous potential [23,24], the local uniqueness and refined spike profiles of ground states for the GP functional are analyzed when $\lambda$ tends to a critical value ${\lambda }^{*}$.

However, as far as we know, there are few papers using the constrained variational approaches to study the varying non-local problem (1). Inspired by the above articles, the aim of the present paper is to study the following constrained minimization problem related to (1), which is defined by

$$I(\eta ,s,\lambda ):=\underset{u\in \mathcal{U}}{inf}E\left(u\right),$$

(2)

where $E\left(u\right)$ fulfills

$$\begin{array}{cc}\hfill E\left(u\right):& =\frac{\eta }{2}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+\frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^3}{V\left(x\right)\left|u\right|}^{2}dx-\frac{\lambda }{p+2}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p+2}dx.\hfill \end{array}$$

(3)

The above $\mathcal{U}$ in (2) is restricted to meet

$$\mathcal{U}:=\left\{u\in \mathcal{H}\mid {\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx=1\right\},$$

(4)

where $\mathcal{H}$ satisfies

$$\mathcal{H}:=\left\{u\in H^1\left({\mathbb{R}}^3\right)\mid {\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u\right|}^{2}dx<\infty \right\}$$

as well as with the norm ${\parallel u\parallel }_{\mathcal{H}}:={\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+{\int }_{{\mathbb{R}}^3}\left(1+{V\left(x\right)\left|u\right|}^2\right)dx\right)}^{\frac{1}{2}}$. Assume that the $V\left(x\right)$ in (1) satisfies

$$\left(V_1\right).V\left(x\right)\in L_{loc}^{\infty }\left({\mathbb{R}}^3\right)\cap C_{loc}^{\alpha }\left({\mathbb{R}}^3\right),\alpha \in (0,1),\underset{\left|x\right|\to \infty }{lim}V\left(x\right)=+\infty \mathrm{and}\underset{{\mathbb{R}}^3}{min}V\left(x\right)=0.$$

To state our main results, we introduce an elliptic equation such as

$$-\frac{3p}{4}\Delta Q_p+\left(1-\frac{p}{4}\right)Q_p-{\left|Q_p\right|}^p{Q}_p=0,x\in {\mathbb{R}}^3,0<p<4.$$

(5)

In fact, up to the translations, (5) has a unique positive radially symmetric solution $Q_p\in H^1\left({\mathbb{R}}^3\right)$, as seen in [25]. For convenience, we denote a critical constant

$${\lambda }^{*}:=\frac{b}{(s+1)}{\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}},$$

where Q is the unique positive solution of (5) for $p=\frac{4(s+1)}{3}$. According to the above conditions, the existence and non-existence theorems on constraint minimizers for $I(\eta ,s,\lambda )$ are established as follows:

Theorem 1.

For $\eta ,s>0$, $0<p<4$ and if $\left(V_1\right)$ holds, then $I(\eta ,s,\lambda )$ has at least one minimizer for $p<\frac{4(s+1)}{3}$ or $p=\frac{4(s+1)}{3},0<\lambda \le {\lambda }^{*}$. The $I(\eta ,s,\lambda )$ has no minimizer for $p>\frac{4(s+1)}{3}$ or $p=\frac{4(s+1)}{3},\lambda >{\lambda }^{*}$.

Theorem 2.

For $\eta =0$, $s>0$, $p=\frac{4(s+1)}{3}$ and if $\left(V_1\right)$ holds, then $I(\eta ,s,\lambda )$ has at least one minimizer if $0<\lambda <{\lambda }^{*}$. Moreover, $I(\eta ,s,\lambda )$ has no minimizer for $\lambda \ge {\lambda }^{*}$

Remark that similar conclusions appear elsewhere for studying different types of Kirchhoff equations, as seen in [7,12,14,15]. For convenience, we give a detailed proof of Theorems 1 and 2 in Section 3. In view of the above theorems, one knows that, for $\eta >0$, $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$, the $I(\eta ,s,\lambda )$ has at least one minimizer. However, for $\eta =0$, $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$, the $I(\eta ,s,{\lambda }^{*})$ admits no minimizer. A nature question is what happens to constraint minimizers of $I(\eta ,s,\lambda )$ when $\eta$ tends to 0 from the right?

Suppose that $u_{\eta }$ is a minimizer for $I(\eta ,s,\lambda )$; then, one can restrict $u_{\eta }\ge 0$ due to $E\left(u\right)\ge E\left(\right|u\left|\right)$ for any $u\in \mathcal{U}$. At the same time, we always assume that $I(\eta ,s,\lambda )$ admits a positive minimizer by applying the strong maximum principle to (1). In truth, for any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, one can verify that the positive constraint minimizers $u_{{\eta }_k}$ satisfy ${\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\to +\infty$ as $k\to \infty$ (see Section 4); that is, the minimizers enact blow-up behavior as ${\eta }_k\to 0^{+}$. In order to obtain a more detailed limit behavior of the constraint minimizers, some appropriate assumptions on $V\left(x\right)$ are necessary. For this purpose, we assume that $V\left(x\right)$ is a form of polynomial function, and admits $n\ge 1$ isolated minima. More narrowly, there exist $n\ge 1$ distinct points $x_i\in {\mathbb{R}}^3$, numbers $q_i>0$ and constant $\mathcal{M}>0$ fulfilling

$$\left(V_2\right).V\left(x\right)=C\left(x\right)\prod _{i=1}^n{|x-x_i|}^{q_i}\mathrm{with}\mathcal{M}<C\left(x\right)<\frac{1}{\mathcal{M}}\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^3;$$

here, $\underset{x\to x_i}{lim}C\left(x\right)$ exists for all $1\le i\le n$. For convenience, we denote

$$q=max\{q_1,\cdots ,q_n\}>0,$$

(6)

$${\theta }_i=\frac{1}{{\parallel Q\parallel }_{L^2}^2}\underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^3}{\left|x\right|}^q{\left|Q\left(x\right)\right|}^{2}dx>0,$$

where $Q\left(x\right)$ satisfies (5) for $p=\frac{4(s+1)}{3}$. Moreover, let

$$\theta =min\{{\theta }_1,\cdots {\theta }_n\}>0$$

(7)

and the set of flattest global minima for $V\left(x\right)$ is denoted by

$$W=\{x_i:{\theta }_i=\theta \}.$$

(8)

In light of Theorems 1 and 2, and inspired by [12,14,15,26], for any positive sequence $\left\{{\eta }_k\right\}$ and set $u_{{\eta }_k}$ being the positive minimizers of $I({\eta }_k,s,{\lambda }^{*})$, we next establish the following theorem on limit behavior of constraint minimizers for $I({\eta }_k,s,{\lambda }^{*})$ when $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$ as ${\eta }_k\to 0^{+}$.

Theorem 3.

Assume that $\left(V_1\right)$ and $\left(V_2\right)$ hold. For $p=\frac{4(s+1)}{3}$, $\lambda ={\lambda }^{*}$ and any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, define ${ϵ}_{{\eta }_k}:={\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^{-\frac{1}{2}}$; then, the following conclusions hold:

(i) 

The $u_{{\eta }_k}$ has a unique local maximum $z_{{\eta }_k}$ satisfying $\underset{k\to \infty }{lim}{z}_{{\eta }_k}=x_i$ and $x_i\in W$ is a flattest global minimum of $V\left(x\right)$. Moreover, we have as $k\to \infty$

$${ϵ}_{{\eta }_k}^{\frac{3}{2}}{u}_{{\eta }_k}({ϵ}_{{\eta }_k}x+z_{{\eta }_k})\to \frac{Q\left(\right|x\left|\right)}{{\parallel Q\parallel }_2}\mathit{strongly}\mathit{in}{H}^1\left({\mathbb{R}}^3\right),$$

where Q denotes the unique positive solution of (5) for $p=\frac{4(s+1)}{3}$.

(ii) 

The ${ϵ}_{{\eta }_k}$ fulfills as $k\to \infty$

$${ϵ}_{{\eta }_k}\approx {\left(q\theta \right)}^{-\frac{1}{q+2}}{\left({\eta }_k\right)}^{\frac{1}{q+2}}.$$

(iii) 

The least energy $I({\eta }_k,s,{\lambda }^{*})$ satisfies as $k\to \infty$

$$I({\eta }_k,s,{\lambda }^{*})\approx [\frac{1}{2}{q}^{\frac{2}{q+2}}+q^{\frac{-q}{q+2}}]{\theta }^{\frac{2}{q+2}}{\left({\eta }_k\right)}^{\frac{q}{q+2}},$$

where $q,\theta$ are stated by (6) and (7).

Notice that the $f\left({\eta }_k\right)\approx g\left({\eta }_k\right)$ in Theorem 3 means $f/g\to 1$ as $k\to \infty$. In fact, for the case in which $s=0$ and $V\left(x\right)$ behave in sinusoidal, ring-shaped, periodic and multi-well forms, the papers [19,20,21,22] widely studied the mass concentration behavior of the constrained minimizers.Particularly for $s=1$, the authors in [14,15] also analyzed the limit behavior of minimizers when $\eta >0$ as $b\to 0^{+}$ or $b>0$ as $\eta \to 0^{+}$. As described in Theorem 3, our paper obtains an interesting result on this topic when it involves a varying non-local term, and it thus enriches the study of such issues.

The present paper is structured as follows. Section 3 shall establish the existence and non-existence proof of constrained minimizers for $I(\eta ,s,\lambda )$ when the parameters $\eta ,\lambda$ and exponents $s,p$ satisfy suitable range. For $p=\frac{4(s+1)}{3}$, $\lambda ={\lambda }^{*}$ and any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, in Section 4 we plan to give the accurate energy estimation of $I({\eta }_k,s,{\lambda }^{*})$, and then analyze the detailed limit behavior of positive constrained minimizers as ${\eta }_k\to 0^{+}$.

2. Preliminaries

In this paper, we shall make full use of the following notations:

• The $H^1\left({\mathbb{R}}^3\right)$ is a Sobolev space with norm ${\parallel u\parallel }_{H^1}=:{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx\right)}^{\frac{1}{2}}$.
• On any compact support set of ${\mathbb{R}}^3$, the $L_{loc}^{\infty }\left({\mathbb{R}}^3\right)$ denotes the essentially bounded measurable function space, and $C_{loc}^{\alpha }\left({\mathbb{R}}^3\right)$ is a Hölder continuous function space.
• The $L^p\left({\mathbb{R}}^3\right)$, $p\in (1,\infty )$ denotes a Sobolev space with norm ${\parallel u\parallel }_{L^p}=:{\left({\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx\right)}^{\frac{1}{p}}$.
• The symbol → (resp. ⇀) means the strong (resp. weak) convergence.
• The letters $\mathcal{A}$, $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$, $\mathcal{F}$, $\mathcal{K}$ and $\mathcal{M}$ represent different positive constants.

Moreover, we introduce the following equality, as seen in [9]:

$$\parallel \nabla Q_p{\parallel }_{L^2}^2=\parallel Q_p{\parallel }_{L^2}^2=\frac{2}{p+2}{\parallel Q_p\parallel }_{L^{p+2}}^{p+2},0<p<4.$$

(9)

Recall also from [27] (Proposition 4.1) that $Q_p\left(x\right)$ has the exponential decay property

$$|\nabla Q_p\left(x\right)|,Q_p\left(\right|x\left|\right)={O\left(\right|x|}^{-1}{e}^{-\left|x\right|}\left)\mathrm{as}\right|x|\to \infty .$$

(10)

At last, we give a Gagliardo–Nirenberg (G-N)-type inequality [28] such as

$${\parallel u\parallel }_{L^{2+p}}^{2+p}\le \frac{p+2}{2\parallel Q_p{\parallel }_{L^2}^p}{\parallel \nabla u\parallel }_{L^2}^{\frac{3p}{2}}{\parallel u\parallel }_{L^2}^{2-\frac{p}{2}},0<p<4,$$

(11)

where $Q_p$ is the unique positive solution of (5).

For proving the existence of constraint minimizers, the following compactness lemma is necessary:

Lemma 4

([29] (Theorem 2.1)). Suppose that $\left(V_1\right)$ is holding; then, for any $p\in (2,6)$, the imbedding

$$\mathcal{H}\hookrightarrow L^p\left({\mathbb{R}}^3\right),$$

is compact, where $\mathcal{H}$ is given by (4).

3. Proof of Theorems 1 and 2

In this section, we shall give the proof of existence and non-existence on constraint minimizers for (2), which are divided into the following two parts:

Proof of Theorem 1.

Under the assumption of Theorem 1, for any $u\in \mathcal{U}$, we deduce from G-N inequality (11) that for $\eta >0$, $p<\frac{4(s+1)}{3}$

$$\begin{array}{cc}\hfill E\left(u\right)& \ge \frac{\eta }{2}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+\frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u\right|}^{2}dx-\frac{\lambda }{2\parallel Q_p{\parallel }_{L^2}^p}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{\frac{3p}{4}}.\hfill \end{array}$$

(12)

For $p=\frac{4(s+1)}{3}$ and $0<\lambda \le {\lambda }^{*}=\frac{b}{(s+1)}{\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}$, one derives from (11) that

$$\begin{array}{cc}\hfill E\left(u\right)& \ge \frac{\eta }{2}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+\frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u\right|}^{2}dx-\frac{\lambda }{2\parallel Q_p{\parallel }_{L^2}^p}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & \ge \frac{\eta }{2}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx+\frac{{\lambda }^{*}-\lambda }{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}+\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u\right|}^{2}dx.\hfill \end{array}$$

(13)

Both $p<\frac{4(s+1)}{3}$ and $p=\frac{4(s+1)}{3},0<\lambda \le {\lambda }^{*}$ hold, (12) and (13) yield a fact that, for any sequence $\left\{u_n\right\}\subseteq \mathcal{U}$, the $E\left(u_n\right)$ is bounded uniformly from below. Hence, there admits a minimization sequence $\left\{u_n\right\}\subseteq \mathcal{U}$ as fulfilling

$$I(\eta ,s,\lambda )=\underset{n\to \infty }{lim}E\left(u_n\right).$$

In truth, one can obtain from (12) and (13) that $\left\{u_n\right\}$ is bounded in $\mathcal{H}$. Applying the Lemma 4, there exists a $\overline{u}\in \mathcal{H}$, and $\left\{u_n\right\}$ has a subsequence $\left\{u_{n_k}\right\}$ such that as $k\to \infty$

$$u_{n_k}\rightharpoonup \overline{u}\mathrm{weakly}\mathrm{in}\mathcal{H},u_{n_k}\to \overline{u}\mathrm{strongly}\mathrm{in}{L}^{\nu }\left({\mathbb{R}}^3\right),2<\nu <6.$$

Using the weak lower semi-continuity, we obtain

$$\underset{k\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^3}|\nabla u_{n_k}{|}^{2}dx\ge {\int }_{{\mathbb{R}}^3}{|\nabla \overline{u}|}^{2}dx.$$

The above results give that

$$I(\eta ,s,\lambda )=\underset{k\to \infty }{lim\; inf}E\left(u_{n_k}\right)\ge E\left(\overline{u}\right)\ge I(\eta ,s,\lambda )$$

which then yields $E\left(\overline{u}\right)=I(\eta ,s,\lambda )$. Hence, $\overline{u}$ is a minimizer for $I(\eta ,s,\lambda )$.

The non-existence proof of constraint minimizer comes true by establishing energy estimation for $I(\eta ,s,\lambda )$. To meet this goal, we choose a test function such as

$$u_t\left(x\right):=\frac{{\mathcal{P}}_t}{{\parallel Q\parallel }_{L^2}}{t}^{\frac{3}{2}}\Phi (x-x_i)Q\left(t\right|x-x_i\left|\right)(t>0),$$

(14)

where Q fulfills (5) for $p=\frac{4(s+1)}{3}$, and $x_i\in W$ satisfies $V\left(x_i\right)=0$. The function $\Phi \left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^3\right)$ in (14) is chosen as

$$\left\{\begin{array}{cc}\hfill \Phi \left(x\right)=1,& \left|x\right|\le 1,\hfill \\ \hfill 0<\Phi \left(x\right)<1,& 1<\left|x\right|<2,\hfill \\ \hfill |\Phi \left(x\right)|=0,& \left|x\right|\ge 2,\hfill \\ \hfill |\nabla \Phi \left(x\right)|\le C,& x\in {\mathbb{R}}^3.\hfill \end{array}\right.$$

Notice that ${\mathcal{P}}_t$ in (14) makes sure $\parallel u_t{\parallel }_{L^2}^2=1$. It is deduced from (10) and (14) that

$$1\le P_t\le 1+O\left(t^{-\infty }\right)\mathrm{and}\underset{t\to +\infty }{lim}{P}_t=1,$$

(15)

where $g\left(t\right)=O\left(t^{-\infty }\right)$ means $\underset{t\to +\infty }{lim}\left|g\left(t\right)\right|t^d=0$ for any $d>0$. One can attain from (9) that as $t\to \infty$

$$\begin{array}{cc}\hfill I(\eta ,s,\lambda )& \le \frac{\eta P_t^2{t}^2}{{2\parallel Q\parallel }_{L^2}^2}{\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx+\frac{b{P}_t^{2(s+1)}{t}^{2(s+1)}}{{2(s+1)\parallel Q\parallel }_{L^2}^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & -\frac{\lambda P_t^{p+2}{t}^{\frac{3p}{2}}}{{(p+2)\parallel Q\parallel }_{L^2}^{p+2}}{\int }_{{\mathbb{R}}^3}{\left|Q\right|}^{p+2}dx+V\left(x_0\right)+o\left(1\right)+O\left(t^{-\infty }\right)\hfill \end{array}$$

(16)

which yields that, for any $p>\frac{4(s+1)}{3}$, the $I(\eta ,s,\lambda )\to -\infty$ as $t\to \infty$. For $\eta >0$ and $p=\frac{4(s+1)}{3}$, we derive from (16) that

$$\begin{array}{cc}\hfill I(\eta ,s,\lambda )& \le \frac{\eta t^2}{2}+\frac{b{t}^{2(s+1)}}{2(s+1)}-\frac{\lambda t^{2(s+1)}}{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}+o\left(1\right)+O\left(t^{-\infty }\right)\hfill \\ \hfill & =\frac{\eta t^2}{2}+\frac{({\lambda }^{*}-\lambda )t^{2(s+1)}}{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}+o\left(1\right)+O\left(t^{-\infty }\right)\to -\infty .\hfill \end{array}$$

(17)

We can deduce from (17) that for $\lambda >{\lambda }^{*}$, the $I(\eta ,s,\lambda )\to -\infty$ as $t\to \infty$. Hence, for any $\eta >0$, if either $p>\frac{4(s+1)}{3}$ or $p=\frac{4(s+1)}{3}$, $\lambda >{\lambda }^{*}$ holds, then $I(\eta ,s,\lambda )$ has no minimizer. □

Proof of Theorem 2.

Under the assumption of Theorem 2, for any $u\in \mathcal{U}$, one can derive from (11) that for $\eta =0$ and $p=\frac{4(s+1)}{3}$ that

$$E\left(u\right)\ge \frac{{\lambda }^{*}-\lambda }{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx\right)}^{s+1}+\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u\right|}^{2}dx.$$

(18)

If $0<\lambda <{\lambda }^{*}$, repeating the proof of Theorem 1, one claims that $I(0,s,\lambda )$ has a minimizer.

The non-existence proof of constraint minimizer is established as follows: for $\eta =0$ and $p=\frac{4(s+1)}{3}$, similar to the estimation of (17), one obtains that

$$I(0,s,\lambda )\le E\left(u_t\right)=\frac{({\lambda }^{*}-\lambda )t^{2(s+1)}}{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}+o\left(1\right)+O\left(t^{-\infty }\right)\to -\infty$$

(19)

It then yields that $I(0,s,\lambda )$ has no minimizer due to $I(0,s,\lambda )=-\infty$ for $\lambda >{\lambda }^{*}$.

For $\eta =0$ and $\lambda ={\lambda }^{*}$, one can obtain from (18) and (19) that $I(0,s,{\lambda }^{*})=0$. We next argue that $I(0,s,{\lambda }^{*})$ has no minimizer by establishing a contradiction. If this is not true, suppose that $\widehat{u}\in \mathcal{U}$ is a minimizer of $I(0,s,{\lambda }^{*})$. As stated in Section 1, we may assume that $\widehat{u}$ is positive. Since $V\left(x\right)\ge 0$ and ${\int }_{{\mathbb{R}}^3}{\left|\widehat{u}\right|}^{2}dx=1$, the G-N inequality (11) then yields that

$$\frac{1}{(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla \widehat{u}|}^{2}dx\right)}^{s+1}=\frac{3{\lambda }^{*}}{2s+5}{\int }_{{\mathbb{R}}^3}{\left|\widehat{u}\right|}^{\frac{4(s+1)}{3}+2}dx,$$

where the equality holds only for $\widehat{u}=Q$, and Q is the unique positive solution of (5) for $p=\frac{4(s+1)}{3}$. One obtains from (12) that $\widehat{u}$ satisfies

$${\int }_{{\mathbb{R}}^3}V\left(x\right){\widehat{u}}^{2}dx=\underset{{\mathbb{R}}^3}{min}V\left(x\right)=0.$$

However, the above two equalities cannot be held at the same time because the first one presents a fact that $\widehat{u}$ has no compact support, and the second one needs $\widehat{u}=Q$ to possess a compact support. Thus, one claims that $I(0,s,{\lambda }^{*})$ has no minimizer. So far, the non-existence proof of constraint minimizer is completed. □

4. Proof of Theorem 3

In this section, for $p=\frac{4(s+1)}{3}$, $\lambda ={\lambda }^{*}$ and any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, we plan to analyze the limit behavior on minimizers $u_{{\eta }_k}$ for $I({\eta }_k,s,{\lambda }^{*})$ as ${\eta }_k\to 0^{+}$. Before proving Theorem 3, some indispensable lemmas are necessary, which are stated as follows:

Lemma 5.

Under the assumption of Theorem 3, set ${\widehat{v}}_{{\eta }_k}\left(x\right):={ϵ}_{{\eta }_k}^{\frac{3}{2}}{u}_{{\eta }_k}\left({ϵ}_{{\eta }_k}x\right)$ and ${ϵ}_{{\eta }_k}=({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^2$$dx{)}^{-\frac{1}{2}}>0$; then, as $k\to \infty$, the ${ϵ}_{{\eta }_k}\to 0$ and ${\widehat{v}}_{{\eta }_k}$ satisfy

$${\int }_{{\mathbb{R}}^3}|\nabla {\widehat{v}}_{{\eta }_k}{|}^{2}dx=1,{\int }_{{\mathbb{R}}^3}{\left|{\widehat{v}}_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}+2}\to \frac{b\left[2s+5\right]}{3(s+1){\lambda }^{*}}.$$

Proof. 

If $u_{{\eta }_k}$ are positive minimizers of (2), then $u_{{\eta }_k}$ satisfies

$$-\left({\eta }_k+b{\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^s\right)\Delta u_{{\eta }_k}+V\left(x\right)u_{{\eta }_k}={\mu }_{{\eta }_k}{u}_{{\eta }_k}+{\lambda }^{*}{\left|u_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}}{u}_{{\eta }_k}$$

(20)

here, ${\mu }_{{\eta }_k}\in \mathbb{R}$ denote Lagrange multipliers. Set

$${\widehat{v}}_{{\eta }_k}\left(x\right):={ϵ}_{{\eta }_k}^{\frac{3}{2}}{u}_{{\eta }_k}\left({ϵ}_{{\eta }_k}x\right),$$

(21)

where ${ϵ}_{{\eta }_k}={\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^{-\frac{1}{2}}>0.$ On the contrary, we assume that ${ϵ}_{{\eta }_k}\nrightarrow 0$ as ${\eta }_k\to 0^{+}$; then, $\left\{u_{{\eta }_k}\right\}$ is bounded uniformly in $\mathcal{H}$. Similar to the proof of Theorems 1 and 2 in Section 3, one asserts that there exists a $u_0\in \mathcal{U}$ and $\left\{u_{{\eta }_k}\right\}$ has a subsequence (still denoted by $\left\{u_{{\eta }_k}\right\}$) such that as ${\eta }_k\to 0^{+}$

$$u_{{\eta }_k}\rightharpoonup u_0\mathrm{weakly}\mathrm{in}\mathcal{H},u_{{\eta }_k}\to u_0\mathrm{strongly}\mathrm{in}{L}^p\left({\mathbb{R}}^3\right),2<p<6.$$

(22)

To obtain our result, we need to prove that $I({\eta }_k,s,{\lambda }^{*})\to 0$ as ${\eta }_k\to 0^{+}$. For this purpose, we choose a test function the same as (14). Based on (10), (14) and (15), one calculates that

$${\int }_{{\mathbb{R}}^3}|\nabla u_t{|}^{2}dx=\frac{P_t^2{t}^2}{{\parallel Q\parallel }_{L^2}^2}{\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx+O\left(t^{-\infty }\right)$$

and

$${\int }_{{\mathbb{R}}^3}|u_t{|}^{\frac{4(s+1)}{3}+2}dx=\frac{P_t^{\frac{4(s+1)}{3}+2}{t}^{2(s+1)}}{{\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}+2}}{\int }_{{\mathbb{R}}^3}{\left|Q\right|}^{\frac{4(s+1)}{3}+2}dx+O\left(t^{-\infty }\right).$$

Since $V\left(x\right)$ satisfies $\left(V_1\right)$ and $\left(V_2\right)$, one obtains that as $t\to \infty$

$${\int }_{{\mathbb{R}}^3}V\left(x\right){\left|u_t\right|}^{2}dx=V\left(x_i\right)+o\left(1\right)=o\left(1\right).$$

It thus follows from (9) that, for $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$ as $t\to +\infty$

$$\begin{array}{cc}\hfill I({\eta }_k,s,{\lambda }^{*})& \le \frac{{\eta }_k{P}_t^2{t}^2}{{2\parallel Q\parallel }_{L^2}^2}{\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx+\frac{b{P}_t^{2(s+1)}{t}^{2(s+1)}}{{2(s+1)\parallel Q\parallel }_{L^2}^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & -\frac{{\lambda }^{*}{P}_t^{p+2}{t}^4}{{(p+2)\parallel Q\parallel }_{L^2}^{p+2}}{\int }_{{\mathbb{R}}^3}{\left|Q\right|}^{p+2}dx+V\left(x_i\right)+o\left(1\right)+O\left(t^{-\infty }\right)\hfill \\ \hfill & =\frac{{\eta }_k{t}^2}{2}+\frac{b{t}^{2(s+1)}}{2(s+1)}-\frac{{\lambda }^{*}{t}^{2(s+1)}}{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}+o\left(1\right)+O\left(t^{-\infty }\right)\hfill \\ \hfill & =\frac{{\eta }_k{t}^2}{2}+o\left(1\right)+O\left(t^{-\infty }\right).\hfill \end{array}$$

(23)

Taking $t={\left({\eta }_k\right)}^{-\frac{1}{3}}$ into (23), it yields that as $k,t\to \infty$

$$I({\eta }_k,s,{\lambda }^{*})\le E\left(u_t\right)=\frac{{\eta }_k^{\frac{1}{3}}}{2}+o\left(1\right)+O\left(t^{-\infty }\right)\to 0.$$

(24)

We can deduce from (3), (22) and (24) that

$$0=I(0,s,{\lambda }^{*})\le E\left(u_0\right)\le \underset{k\to \infty }{lim\; inf}E\left(u_{{\eta }_k}\right)=\underset{k\to \infty }{lim}I({\eta }_k,s,{\lambda }^{*})=I(0,s,{\lambda }^{*})=0$$

which yields a fact that $u_0$ is a minimizer of $I(0,s,{\lambda }^{*})$. However, this is a contradiction since Theorem 2 shows that $I(0,s,{\lambda }^{*})$ has no minimizer. Thus, ${ϵ}_{{\eta }_k}\to 0$ holds as $k\to \infty$.

By (21), we just have ${\int }_{{\mathbb{R}}^3}|\nabla {\widehat{v}}_{{\eta }_k}{|}^{2}dx={ϵ}_{{\eta }_k}^{-2}{\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx=1.$ Since $u_{{\eta }_k}$ are minimizers of $I({\eta }_k,s,{\lambda }^{*})$ for any ${\eta }_k>0$, we can derive from (11) and (24) that as $k\to \infty$

$$0\le E\left(u_{{\eta }_k}\right)=I({\eta }_k,s,{\lambda }^{*})\le E\left(u_t\right)\to 0$$

which yields that as $k\to \infty$

$$\frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^{2(s+1)}-\frac{3{\lambda }^{*}}{4s+10}{\int }_{{\mathbb{R}}^3}{\left|u_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}+2}dx\to 0.$$

(25)

It hence follows from (21) and (25) that as $k\to \infty$

$$\frac{b}{2(s+1)}-\frac{3{\lambda }^{*}}{4s+10}{\int }_{{\mathbb{R}}^3}{\left|{\widehat{v}}_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}+2}dx\to 0,$$

which shows as $k\to \infty$

$${\int }_{{\mathbb{R}}^3}{\left|{\widehat{v}}_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}+2}dx\to \frac{b\left[2s+5\right]}{3(s+1){\lambda }^{*}}.$$

We have finished the proof of Lemma 5. □

Assume that $u_{{\eta }_k}$ are positive minimizers of $I({\eta }_k,s,{\lambda }^{*})$ for any ${\eta }_k>0$. Since ${\int }_{{\mathbb{R}}^3}{\left|u_{{\eta }_k}\right|}^2$$dx=1$, one has $u_{{\eta }_k}\left(x\right)\to 0$ as $\left|x\right|\to \infty$. This yields that $u_{{\eta }_k}\left(x\right)$ has at least one local maximum, which is denoted by $z_{{\eta }_k}$. We define a function

$$v_{{\eta }_k}\left(x\right):={ϵ}_{{\eta }_k}^{\frac{3}{2}}{u}_{{\eta }_k}({ϵ}_{{\eta }_k}x+z_{{\eta }_k}),$$

(26)

where ${ϵ}_{{\eta }_k}$ is given in Lemma 5. We next establish the following lemma, which is related to convergence properties of $v_{{\eta }_k}$ and $z_{{\eta }_k}$.

Lemma 6.

Under the assumption of Theorem 3, set $z_{{\eta }_k}$ as a local maximum of $u_{{\eta }_k}$ and $v_{{\eta }_k}$ defined by (26); then, we have

(i) 

There exist a finite ball $B_{2s}\left(0\right)$ and a constant $\mathcal{D}>0$ such that

$$\underset{k\to \infty }{lim\; inf}{\int }_{B_{2s}\left(0\right)}{\left|v_{{\eta }_k}\left(x\right)\right|}^{2}dx\ge \mathcal{D}>0.$$

(ii) 

The $z_{{\eta }_k}$ is a unique maximum of $u_{{\eta }_k}$ and satisfies $z_{{\eta }_k}\to x_0$ for some $x_0\in {\mathbb{R}}^3$ as $k\to \infty$. Furthermore, the $x_0$ is a minimum of $V\left(x\right)$, that is, $V\left(x_0\right)=0$.

(iii) 

The function $v_{{\eta }_k}$ satisfies

$$\underset{k\to \infty }{lim}{v}_{{\eta }_k}\left(x\right)=\underset{k\to \infty }{lim}{ϵ}_{{\eta }_k}^{\frac{3}{2}}{u}_{{\eta }_k}({ϵ}_{{\eta }_k}x+z_{{\eta }_k})=\frac{Q\left(\right|x\left|\right)}{{\parallel Q\parallel }_{L^2}}\mathit{strongly}\mathit{in}{H}^1\left({\mathbb{R}}^3\right),$$

where Q is the unique solution of (5) for $p=\frac{4(s+1)}{3}$.

Proof. 

(i) By (20), we see that $v_{{\eta }_k}$ fulfills the elliptic equation

$$-\left({\eta }_k{ϵ}_{{\eta }_k}^{2s}+b\right)\Delta v_{{\eta }_k}+{ϵ}_{{\eta }_k}^{2(s+1)}V\left(x\right)v_{{\eta }_k}={ϵ}_{{\eta }_k}^{2(s+1)}{\mu }_{{\eta }_k}{v}_{{\eta }_k}+{\lambda }^{*}{\left|v_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}}{v}_{{\eta }_k},$$

(27)

here, ${\mu }_{{\eta }_k}$ are Lagrange multipliers. In truth, (2) and (20) give that

$$\begin{array}{cc}\hfill {\mu }_{{\eta }_k}& =2I({\eta }_k,s,{\lambda }^{*})+\frac{sb}{s+1}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^{s+1}-\frac{2(s+1){\lambda }^{*}}{2s+5}{\int }_{{\mathbb{R}}^3}{\left|u_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}+2}dx.\hfill \end{array}$$

(28)

Repeating the proof of (24), one obtains that as $k\to \infty$

$${ϵ}_{{\eta }_k}^{2(s+1)}I({\eta }_k,s,{\lambda }^{*})\to 0\mathrm{and}{\int }_{{\mathbb{R}}^3}V({ϵ}_{{\eta }_k}x+z_{{\eta }_k})v_{{\eta }_k}^2\left(x\right)dx\to 0.$$

(29)

Since $0<p=\frac{4(s+1)}{3}<4$ yields $0<s<2$, we can obtain from (28), (29) and Lemma 5 that as $k\to \infty$

$${\mu }_{{\eta }_k}{ϵ}_{{\eta }_k}^{2(s+1)}=2{ϵ}_{{\eta }_k}^{2(s+1)}I({\eta }_k,s,{\lambda }^{*})+\frac{sb}{s+1}-\frac{2b}{3}\to \frac{(s-2)b}{3(s+1)}<0.$$

(30)

Since $u_{{\eta }_k}$ take local maxima at $x=z_{{\eta }_k}$, it yields that $v_{{\eta }_k}$ obtain local maxima at $x=0$. We thus derive from (27) and (30) that there exists a constant $\mathcal{K}>0$ satisfying as $k\to \infty$

$$v_{{\eta }_k}\left(0\right)\ge \mathcal{K}>0.$$

(31)

Furthermore, one obtains from (27) that

$$-\Delta v_{{\eta }_k}-c\left(x\right)v_{{\eta }_k}\le 0,x\in {\mathbb{R}}^3,$$

(32)

where $c\left(x\right)={\lambda }^{*}{\left|v_{{\eta }_k}\right|}^{\frac{4(s+1)}{3}}$. In view of the De Giorgi–Nash–Moser theory, as seen in [30] (Theorem 4.1), one declares that there exist a finite ball $B_{2s}\left(0\right)\subset {\mathbb{R}}^3$ and constant $\mathcal{C}>0$ such that

$$\underset{B_s\left(0\right)}{max}{v}_{{\eta }_k}\le \mathcal{C}{\left({\int }_{B_{2s}\left(0\right)}{\left|v_{{\eta }_k}\right|}^{2}dx\right)}^{\frac{1}{2}}.$$

(33)

It hence yields from (31) and (33) that there exists a constant $\mathcal{D}>0$ satisfying

$$\underset{k\to \infty }{lim\; inf}{\int }_{B_{2s}\left(0\right)}{\left|v_{{\eta }_k}\right|}^{2}dx\ge \mathcal{D}>0.$$

(34)

(ii) On the contrary, one may assume that $|z_{{\eta }_k}|\to \infty$ as $k\to \infty$. By applying (34) and Fatou’s lemma, for any large constant $\mathcal{A}$, one has

$$\begin{gathered} \underset{k\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^3}V({ϵ}_{{\eta }_k}x+z_{{\eta }_k})|v_{{\eta }_k}\left(x\right){|}^{2}dx \\ \ge {\int }_{B_{2s}\left(0\right)}\underset{k\to \infty }{lim\; inf}V({ϵ}_{{\eta }_k}x+z_{{\eta }_k}){\left|v_{{\eta }_k}\left(x\right)\right|}^{2}dx\ge \mathcal{A}>0 \end{gathered}$$

(35)

which contradicts (29), and it hence shows that $|z_{{\eta }_k}|$ is bounded in ${\mathbb{R}}^3$. Taking a subsequence of $\left\{z_{{\eta }_k}\right\}$ if necessary (still denoted by $\left\{z_{{\eta }_k}\right\}$), there admits a $x_0\in {\mathbb{R}}^3$ such that $z_{{\eta }_k}\to x_0$ as $k\to \infty$. In fact, one can claim that $x_0$ is a minimum of $V\left(x\right)$, that is, $V\left(x_0\right)=0$. If not, repeating the proof of (35), it also yields a contradiction. Thus, we say that $z_{{\eta }_k}\to x_0$ as $k\to \infty$ and $V\left(x_0\right)=0$.

(iii) The Lemma 5 shows that sequence $\left\{v_{{\eta }_k}\right\}$ is bounded in $H^1\left({\mathbb{R}}^3\right)$, and under the sense of subsequence, there exists a $v_0\in H^1\left({\mathbb{R}}^3\right)$ such that $v_{{\eta }_k}\rightharpoonup v_0$ as $k\to \infty$. Using (30) and passing weak limit to (27), one obtains that $v_0$ satisfies

$$-\Delta v_0+\frac{2-s}{3(s+1)}{v}_0=\frac{{\lambda }^{*}}{b}{\left|v_0\right|}^{\frac{4(s+1)}{3}}{v}_0,x\in {\mathbb{R}}^3,$$

(36)

where $0<s<2$. By (34) and applying the strong maximum principle to (36), one has $v_0>0$. Taking $p=\frac{4(s+1)}{3}$ in (5), one knows that

$$-\Delta Q+\frac{2-s}{3(s+1)}Q=\frac{1}{s+1}{\left|Q\right|}^{\frac{4(s+1)}{3}}Q,x\in {\mathbb{R}}^3.$$

(37)

Due to the fact that (37) has a unique positive radially symmetric solution $Q\in H^1\left({\mathbb{R}}^3\right)$, it hence yields from (36) that

$$v_0\left(x\right)=\frac{Q\left(\right|x-y_0\left|\right)}{{\parallel Q\parallel }_{L^2}}\mathrm{for}\mathrm{some}{y}_0\in {\mathbb{R}}^3.$$

(38)

Similar to the procedure of Theorem 1, one declares that as $k\to \infty$, $v_{{\eta }_k}\to v_0$ strongly in $H^1\left({\mathbb{R}}^3\right)$. Using the standard elliptic regularity theory, we obtain from (27) that as $k\to \infty$

$$v_{{\eta }_k}\to v_0\mathrm{in}{C}_{loc}^{2,\alpha }\left({\mathbb{R}}^3\right),\alpha \in (0,1).$$

(39)

Applying the method [18] (Theorem 2), one knows that the $y_0=0$ in (38), and 0 is the unique global maximum of $v_0$. Therefore, $v_0$ behaves like

$$v_0\left(x\right)=\frac{Q\left(\right|x\left|\right)}{{\parallel Q\parallel }_{L^2}},x\in {\mathbb{R}}^3.$$

(40)

By (39), using the technique of [19] (Theorem 1.2), we know that $z_{{\eta }_k}$ is the unique global maximum of $u_{{\eta }_k}$. □

To obtain a more detailed description on limit behavior of constraint minimizers $u_{{\eta }_k}$ as ${\eta }_k\to 0^{+}$, some precise energy estimation of $I({\eta }_k,s,{\lambda }^{*})$ as ${\eta }_k\to 0^{+}$ is necessary. Toward this aim, we begin with the upper-bound estimation of $I({\eta }_k,s,\lambda )$, which is sated as the following lemma:

Lemma 7.

Assume that $\left(V_1\right)$ and $\left(V_2\right)$ hold. If $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$, then for any positive sequence $\left\{{\eta }_k\right\}$ with ${\eta }_k\to 0^{+}$ as $k\to \infty$, the $I({\eta }_k,s,\lambda )$ satisfies as $k\to \infty$

$$I({\eta }_k,s,{\lambda }^{*})\le \left[\frac{1}{2}{q}^{\frac{2}{q+2}}+q^{\frac{-q}{q+2}}\right]{\theta }^{\frac{2}{q+2}}{\left({\eta }_k\right)}^{\frac{q}{q+2}}(1+o\left(1\right)),$$

where $q,\theta$ are defined by (6) and (7).

Proof. 

Choosing (14), we can deduce from (9)–(11) that there exist positive constants $d_1,d_2$ such that as $t\to +\infty$

$$\begin{array}{cc}\hfill & \frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u_t|}^{2}dx\right)}^{s+1}-\frac{3{\lambda }^{*}}{4s+10}{\int }_{{\mathbb{R}}^3}{\left|u_t\right|}^{\frac{4(s+1)}{3}+2}dx\hfill \\ \hfill \le & \frac{b{t}^{2(s+1)}}{2(s+1)}-\frac{{\lambda }^{*}{t}^{2(s+1)}}{{2\parallel Q\parallel }_{L^2}^{\frac{4(s+1)}{3}}}+d_1{e}^{-d_{2}t}=d_1{e}^{-d_{2}t}\hfill \end{array}$$

(41)

and there exist positive constants $d_3,d_4$ such that as $t\to +\infty$

$$\frac{{\eta }_k}{2}{\int }_{{\mathbb{R}}^3}|\nabla u_t{|}^{2}dx=\frac{P_t^2{t}^2}{{\parallel Q\parallel }_{L^2}^2}{\int }_{{\mathbb{R}}^3}{|\nabla Q|}^{2}dx=\frac{{\eta }_k{t}^2}{2}+d_3{e}^{-d_{4}t}.$$

(42)

Since $V\left(x\right)$ satisfies $\left(V_1\right)$ and $\left(V_2\right)$, we derive that there exist positive constants $d_5,d_6$ such that as $t\to +\infty$

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}V\left(x\right)u_t^{2}dx\le \frac{1}{{\parallel Q\parallel }_{L^2}^2}{\int }_{B_{\sqrt{t}}\left(0\right)}V(\frac{x}{t}+x_i){\left|Q\right|}^{2}dx+d_5{e}^{-d_{6}t}\hfill \\ \hfill =& \frac{1}{{\parallel Q\parallel }_{L^2}^2}{\int }_{B_{\sqrt{t}}\left(0\right)}C(\frac{x}{t}+x_i)\prod _{j=1}^n|\frac{x}{t}+x_i-x_j{|}^{q_j}{\left|Q\right|}^{2}dx+d_5{e}^{-d_{6}t}\hfill \\ \hfill =& t^{-q}\frac{1}{{\parallel Q\parallel }_{L^2}^2}\underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^3}{\left|x\right|}^q{\left|Q\left(x\right)\right|}^{2}dx+o\left(t^{-q}\right)+d_5{e}^{-d_{6}t}\hfill \\ \hfill =& \theta t^{-q}+o\left(t^{-q}\right)+d_5{e}^{-d_{6}t},\hfill \end{array}$$

(43)

where $q,\theta$ defined by (6) and (7). Using (41)–(43), we have

$$\begin{array}{cc}\hfill I({\eta }_k,s,{\lambda }^{*})\le & \frac{{\eta }_k{t}^2}{2}+\theta t^{-q}+o\left(t^{-q}\right)+d_1{e}^{-d_{2}t}+d_3{e}^{-d_{4}t}+d_5{e}^{-d_{6}t}\hfill \\ \hfill =& \frac{{\eta }_k{t}^2}{2}+\theta t^{-q}\left(1+o\left(1\right)\right).\hfill \end{array}$$

(44)

Taking $t={\left(q\theta \right)}^{\frac{1}{q+2}}{\left({\eta }_k\right)}^{-\frac{1}{q+2}}$, one can deduce from (44) that as ${\eta }_k\to 0^{+}$

$$I({\eta }_k,s,{\lambda }^{*})\le \left[\frac{1}{2}{q}^{\frac{2}{q+2}}+q^{\frac{-q}{q+2}}\right]{\theta }^{\frac{2}{q+2}}{\left({\eta }_k\right)}^{\frac{q}{q+2}}(1+o\left(1\right)).$$

□

Proof of Theorem 3.

According to the results of Lemmas 5–7, it remains to prove (ii) and (iii) in Theorem 3, which can be realized by establishing the precise lower energy estimation of $I({\eta }_k,s,{\lambda }^{*})$ as ${\eta }_k\to 0^{+}$. To meet this goal, we set $\left\{u_{{\eta }_k}\right\}$ as the positive minimizers of $I({\eta }_k,s,{\lambda }^{*})$, $z_{{\eta }_k}$ being their unique global maxima, and we define $v_{{\eta }_k}$ by (26). Using Lemma 6, one knows that for $\left\{z_{{\eta }_k}\right\}$, choosing a subsequence if necessary (still stated by $\left\{z_{{\eta }_k}\right\}$), the $z_{{\eta }_k}\to x_0$ and $V\left(x_0\right)=0$.

In fact, we can go a step further, that is, we can come to the following conclusion:

$$z_{{\eta }_k}\to x_i\mathrm{and}\frac{|z_{{\eta }_k}-x_i|}{{ϵ}_{{\eta }_k}}\mathrm{is}\mathrm{bounded}\mathrm{uniformly}\mathrm{as}k\to \infty ,$$

(45)

where $x_i\in W$ and $x_i$ denotes a flattest global minimum of $V\left(x\right)$. To obtain (45), we firstly claim that

$$\frac{|z_{{\eta }_k}-x_0|}{{ϵ}_{{\eta }_k}}\mathrm{is}\mathrm{bounded}\mathrm{uniformly}\mathrm{as}k\to \infty .$$

(46)

If this is false, then we assume that $\frac{|z_{{\eta }_k}-x_0|}{{ϵ}_{{\eta }_k}}\to \infty$ as $k\to \infty$. It then follows from $\left(V_2\right)$ and Lemma 6 (i) that, for any large positive constant $\mathcal{F}$,

$$\begin{array}{cc}\hfill & \underset{k\to \infty }{lim\; inf}\frac{1}{{ϵ}_{{\eta }_k}^{q_{i_0}}}{\int }_{{\mathbb{R}}^3}V({ϵ}_{{\eta }_k}x+z_{{\eta }_k})v_{{\eta }_k}^{2}dx\hfill \\ \hfill \ge & C{\int }_{B_{2s}\left(0\right)}\underset{k\to \infty }{lim\; inf}|x+\frac{z_{{\eta }_k}-x_0}{{ϵ}_{{\eta }_k}}{|}^{q_{i_0}}·\prod _{j=1,j\ne i_0}^n{|{ϵ}_{{\eta }_k}x+z_{{\eta }_k}-x_j|}^{q_j}{v}_{{\eta }_k}^{2}dx\ge \mathcal{F}.\hfill \end{array}$$

(47)

Recall from G-N inequality (11) that we also have for $p=\frac{4(s+1)}{3}$ and $\lambda ={\lambda }^{*}$

$$\underset{k\to \infty }{lim\; inf}\left(\frac{b}{2(s+1)}{\left({\int }_{{\mathbb{R}}^3}{|\nabla u_{{\eta }_k}|}^{2}dx\right)}^{s+1}-\frac{{\lambda }^{*}}{p+2}{\int }_{{\mathbb{R}}^3}{\left|u_{{\eta }_k}\right|}^{p+2}dx\right)\ge 0$$

(48)

which together with (47) then gives

$$\underset{k\to \infty }{lim\; inf}I({\eta }_k,s,{\lambda }^{*})=\underset{k\to \infty }{lim\; inf}E\left({\eta }_k\right)\ge \frac{{\eta }_k{ϵ}_{{\eta }_k}^{-2}}{2}+\mathcal{D}{ϵ}_{{\eta }_k}^{q_{i_0}}\ge \mathcal{E}{\eta }_k^{\frac{q_{i_0}}{q_{i_0}+2}},$$

(49)

where $\mathcal{E}$ is a arbitrarily large constant. However, this is a contradiction with the upper energy in Lemma 7. Hence, (46) holds. In truth, the upper energy of $I({\eta }_k,s,{\lambda }^{*})$ also compels that $x_0=x_i\in W$. If not, by repeating the proof process from (46) to (48), one still derives a contradiction. Thus, we complete the proof of (i) in Theorem 3.

Using (45) and similar to estimation of (47), one can deduce that there admits a $\widehat{x}\in {\mathbb{R}}^3$ such that

$$\begin{array}{ll}& {\displaystyle \underset{k\to \infty }{lim\; inf}}\frac{1}{{ϵ}_{{\eta }_k}^q}{\int }_{{\mathbb{R}}^3}V({ϵ}_{{\eta }_k}x+z_{{\eta }_k})v_{{\eta }_k}^{2}dx\\ =& \underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^3}|x+\widehat{x}{|}^q{v}_0^{2}dx\\ \ge & \underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^3}{\left|x\right|}^q{v}_0^{2}dx=\theta ,\end{array}$$

(50)

where $\theta ,q$ given by (6) and (7). As a fact, the equality in (50) holds only for $\overline{x}=0$. One then calculates from (49) and (50) that

$$\underset{k\to \infty }{lim\; inf}I({\eta }_k,s,{\lambda }^{*})=\underset{k\to \infty }{lim\; inf}E\left({\eta }_k\right)\ge \frac{{\eta }_k{ϵ}_{{\eta }_k}^{-2}}{2}+\theta {ϵ}_{{\eta }_k}^q.$$

(51)

Due to the restriction of energy upper bound in Lemma 7, it yields that ${ϵ}_{{\lambda }_k}$ is in the form of

$${ϵ}_{{\eta }_k}={\left(q\theta \right)}^{-\frac{1}{q+2}}{\left({\eta }_k\right)}^{\frac{1}{q+2}}$$

which shows a fact that the (ii) in Theorem 3 holds.

Taking the above ${ϵ}_{{\eta }_k}$ into (51), we can obtain that

$$\underset{k\to \infty }{lim\; inf}I({\eta }_k,s,{\lambda }^{*})\ge \left[\frac{1}{2}{q}^{\frac{2}{q+2}}+q^{\frac{-q}{q+2}}\right]{\theta }^{\frac{2}{q+2}}{\left({\eta }_k\right)}^{\frac{q}{q+2}}.$$

which together with Lemma 7 yields that as $k\to \infty$

$$I({\eta }_k,s,{\lambda }^{*})\approx \left[\frac{1}{2}{q}^{\frac{2}{q+2}}+q^{\frac{-q}{q+2}}\right]{\theta }^{\frac{2}{q+2}}{\left({\eta }_k\right)}^{\frac{q}{q+2}}.$$

So far, we have finished the proof of (iii) in Theorem 3. □

5. Conclusions

There are many significant results for (2) when the exponent $s=1$, and the readers are advised to refer to Section 1. In the present paper, we have studied the constrained minimization problem (2) with $s>0$, which may be the first one studying the varying non-local problem by applying constrained variational methods. Under the assumptions of $\left(V_1\right)$ and $\left(V_2\right)$, our first conclusion is involved in the existence and non-existence of constraint minimizers for (2), which can be stated by Theorems 1 and 2. Furthermore, the second conclusion in Theorem 3 is concerned with the limit behavior of constraint minimizers as ${\eta }_k\to 0^{+}$. In detail, when the trapping potential $V\left(x\right)$ is a polynomial function and fulfills $\left(V_1\right)$ and $\left(V_2\right)$, we can prove that the mass of minimizers must concentrate (i.e., blow up) at some flattest global minimum of $V\left(x\right)$ as ${\eta }_k\to 0^{+}$. However, the local uniqueness of the constraint minimizer is hard to prove as ${\eta }_k\to 0^{+}$. Hence, in the future, we may try to overcome this problem.