Blow-Up Analysis of L²-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term

Abstract

This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of $L^2$-norm solutions for the related equation when the potential function $V\left(x\right)$ fulfills an appropriate choice.

1. Introduction and Main Results

We consider the following elliptic equation with a varying nonlocal term, as follows:

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

(1)

where dimension $n\le 3$, and parameter $\lambda >0$. The exponents $s,p$ in (1) satisfy $s\ge 0$, $0<p<2^{\ast }-2$, where $2^{\ast }=+\infty$ if $n=1,2$ and $2^{\ast }=6$ if $n=3$. The coefficient $\mu$ in (1) is a suitable Lagrange multiplier. The $V\left(x\right)\ge 0$ in (1) is a real function, and it can be regarded as a trapping potential from the physical aspect.

In previous decades, both in the research of mathematical theory and concrete real-world applications, nonlocal problems have attracted a lot of attention. To our knowledge, the most popular nonlocal problem is the following Kirchhoff equation

$$\{\begin{array}{cc}-M\left({\parallel \nabla u\parallel }^p\right){\Delta }_{p}u=f(x,u),\hfill & \hfill x\in \mathsf{\Omega }\\ u=0,\hfill & \hfill x\in \partial \mathsf{\Omega }\end{array}$$

which is related to mechanical phenomena (see [1,2,3,4]). The following nonlocal model,

$$\{\begin{array}{cc}-\Delta u=f(x,u){\left({\int }_{\mathsf{\Omega }}g(x,u)dx\right)}^r,\hfill & \hfill x\in \mathsf{\Omega }\\ u=0,\hfill & \hfill x\in \partial \mathsf{\Omega }\end{array}$$

is widely researched and applied in various areas, such as elastic string theory, population dynamics, plasma physics, heat conduction, etc. The reader may refer to [5,6,7,8] and their references. In recent years, many mathematicians began to study the different types of bi-nonlocal problems similar to the following:

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

which have been deeply studied by applying the variational methods and analytical skills; see [9,10,11,12,13] and the references therein.

The previous works enlighten us to study $L^2$-norm solutions for partial differential Equation (1) with a varying nonlocal term, which can be realized by analyzing the following constrained minimization problem:

$$I(s,p,\lambda ):=\underset{u\in \mathcal{S}}{inf}F\left(u\right),$$

(2)

where $F\left(u\right)$ is an energy functional satisfying the following:

$$F\left(u\right):=\frac{1}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u^{2}dx-\frac{2\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|u\right|}^{p+2}dx.$$

(3)

Note that $\mathcal{S}$ in (2) fulfills the following:

$$\mathcal{S}:=\left\{u\in \mathcal{H},{\int }_{{\mathbb{R}}^n}{\left|u\right|}^2=1\right\}$$

(4)

and $\mathcal{H}$ is defined as follows:

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

with norm ${\parallel u\parallel }_{\mathcal{H}}:={\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx+{\int }_{{\mathbb{R}}^n}\left(1+{V\left(x\right)\left|u\right|}^2\right)dx\right)}^{\frac{1}{2}}$. A well-known result is that the constraint minimizers of (2) can be converted into $L^2$-norm solutions of (1); hence, in the present paper, we only study the constrained minimization problem (2) replacing elliptic Equation (1).

Coincidentally, different forms of constraint minimization problems similar to (2) have been extensively studied. Examples for $s=0,p=2$, and $\lambda >0$ in (1), and $F\left(u\right)$ in (3) are called Gross–Pitaevskii [14,15] energy functionals, which are associated with the attractive Bose–Einstein condensates [16,17]. Many mathematicians are devoted to studying the existence, non-existence, mass concentration behavior, and local uniqueness of solutions when trapping potentials take the forms of polynomial, ring-shaped, multi-well, periodic, and sinusoidal functions; see [18,19,20,21,22,23,24]. Also, for $n\le 4$, $s=1$, and $\lambda >0$, (2) is related to a Kirchhoff-type constraint minimization problem, which has attracted considerable attention from researchers analyzing the existence, local uniqueness, and blow-up behavior of minimizers. Specifically, when the potential $V\left(X\right)=0$, Ye [25,26] obtained the existence and nonexistence results for $L^2$-norm solutions. Zeng and Zhang in [27] gave the local uniqueness of $L^2$-norm solutions, and then in [28], displayed the asymptotic behavior of solutions when the potential satisfied periodic form. In [29], Guo, Zhang, and Zhou analyzed the existence and limit behaviors of $L^2$-norm solutions when the trapping potential satisfied ${lim\; inf}_{\left|x\right|\to +\infty }V\left(x\right)=\infty$. In paper [30,31,32], the authors studied the existence and nonexistence of constraint minimizers for the Kirchhoff-type energy functional with an $L^2$-subcritical term. Also, when the potential was a homogeneous function, Hu and Tang [33] obtained the limit behavior and local uniqueness of $L^2$-norm solutions.

Inspired by the methods and techniques in the above articles, the present paper is mainly concerned with the existence, non-existence, and blow-up behaviors of constraint minimizers for (2). To analyze the existence and non-existence of minimizers, we first assume that $V\left(x\right)$ in (1) satisfies the following restriction:

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

We next introduce the following elliptic equation (see [34]):

$$-\frac{np}{4}\Delta w+(1+\frac{p}{4}(2-n))w-{\left|w\right|}^{p}w=0,x\in {\mathbb{R}}^n,n\le 3,0<p<2^{\ast }-2,$$

(5)

where (5) admits a unique (up to the translations) positive radially symmetric solution $w_p\in H^1\left({\mathbb{R}}^3\right)$. Applying the Pokhozhaev identity to (5), one can obtain the following:

$$\parallel \nabla w_p{\parallel }_{L^2}^2=\parallel w_p{\parallel }_{L^2}^2=\frac{2}{p+2}{\parallel w_p\parallel }_{L^{p+2}}^{p+2}.$$

(6)

A well-known conclusion from ([35], Proposition 4.1) shows that $w_p\left(x\right)$ satisfies the following:

$$|\nabla w_p\left(x\right)|,w_p\left(\right|x\left|\right)={O\left(\right|x|}^{-\frac{n-1}{2}}{e}^{-\left|x\right|}\left)\mathrm{as}\right|x|\to \infty .$$

(7)

A Gagliardo–Nirenberg-type inequality mentioned in [36] is also necessary, such as the following:

$${\parallel u\parallel }_{L^{2+p}}^{2+p}\le c^{\ast }{\parallel \nabla u\parallel }_{L^2}^{\frac{np}{2}}{\parallel u\parallel }_{L^2}^{2+p-\frac{np}{2}},n\le 3,0<p<2^{\ast }-2,$$

(8)

where the best constant $c^{\ast }:=\frac{p+2}{2\parallel w_p{\parallel }_{L^2}^p}$ and $w_p$ is given by (5). Notice that the above equality holds only for $u=w_p$ (up to rescaling).

In truth, the existence and non-existence of constraint minimizers for (2) depend heavily on the exponents $s,p$, and parameter $\lambda$. Thus, we divide the $s,p,\lambda$ into the following cases for convenience.

(c1)

$p<min\{\frac{4(s+1)}{3},4\}$ for $n=3$ and $p\in (0,\frac{4(s+1)}{n})$ for $n=1,2$;

(c2)

$\frac{4(s+1)}{3}<p<4$ for $n=3$ and $\frac{4(s+1)}{n}<p<\infty$ for $n=1,2$;

(c3)

$p=\frac{4(s+1)}{3}<4$ for $n=3$, $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$ and $\lambda \ge {\lambda }^{\ast }:=\frac{\parallel w_p{\parallel }_{L^2}^p}{s+1}$ where $w_p$ is given by (5);

(c4)

$p=\frac{4(s+1)}{3}<4$ for $n=3$, $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, and $0<\lambda <{\lambda }^{\ast }$.

According to the above knowledge points, we give the following existence and nonexistence results of constraint minimizers.

Theorem 1.

Under the assumption $\left(V_1\right)$, if either ($c_1$) or ($c_4$) holds, then (2) has at least one minimizer; if either ($c_2$) or ($c_3$) holds, then (2) has no minimizer. Furthermore, if $p=\frac{4(s+1)}{3}<4$ for $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, we have ${lim}_{\lambda \nearrow {\lambda }^{\ast }}I(s,p,\lambda )=I(s,p,{\lambda }^{\ast })=0$.

For $p=\frac{4(s+1)}{3}<4$ when $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, we can verify that the minimizers $u_{\lambda }$ satisfy ${\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx\to +\infty$ as $\lambda$ tends to ${\lambda }^{\ast }$ from below (see Section 3). This indicates that the minimizers exhibit blow-up behavior as $\lambda \nearrow {\lambda }^{\ast }$. In order to obtain an accurate blow-up rate and the location of the minimizers, we need to make an appropriate assumption about the potential $V\left(x\right)$, such as $V\left(x\right)$ behaving like a polynomial function and having $n\ge 1$ isolated minima. To be more precise, there exist $n\ge 1$ distinct points, $x_i\in {\mathbb{R}}^n$, numbers, $q_i>0$, and positive constants, $0<m<1$ fulfilling the following:

$$V\left(x\right)=h\left(x\right)\prod _{i=1}^n{|x-x_i|}^{q_i}\left(V_2\right)$$

and

$$m<h\left(x\right)<\frac{1}{m}\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^n,$$

(9)

where $\underset{x\to x_i}{lim}h\left(x\right)$ exists for any $1\le i\le n$. For convenience, we present some notations, as follows:

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

(10)

and

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

(11)

Set

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

(12)

and

$$\mathcal{Z}=\{x_i:{\alpha }_i=\alpha \}$$

(13)

denotes a set of the flattest global minima of potential $V\left(x\right)$.

Stimulated by the techniques in [23,32,37], we find that the exact blow-up analysis of minimizers is dependent on the energy of $I(s,p,\lambda )$. Hence, in the following, we establish a refined energy estimation of $I(s,p,{\lambda }_k)$ with any sequence $\left\{{\lambda }_k\right\}$ as ${\lambda }_k\nearrow {\lambda }^{\ast }$.

Theorem 2.

Under the assumptions $\left(V_1\right)$ and $\left(V_2\right)$, if $p=\frac{4(s+1)}{3}<4$ for $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, then for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, the least energy $I(s,p,{\lambda }_k)$ satisfies the following:

$$I(s,p,{\lambda }_k)\approx \left[\frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}}\right]({\lambda }^{\ast }-{\lambda }_k{)}^{\frac{q}{2(s+1)+q}},$$

(14)

where $q,\alpha$ satisfy (10) and (12). Note that $f\left({\lambda }_k\right)\approx g\left({\lambda }_k\right)$ means $f/g\to 1$ as ${\lambda }_k\nearrow {\lambda }^{\ast }$.

When the potential $V\left(x\right)$ in (3) takes forms such as logarithmic, ring-shaped, multi-well, or periodic (see [18,20,21,22,24]), the energy estimation techniques discussed in our article can also address these scenarios. As mentioned in Section 1, the present paper is concerned with the elliptic Equation (1) with a varying nonlocal term, and thus our results extend the corresponding knowledge (see Section 3).

According to Theorem 2, our last result is devoted to studying the blow-up behavior of constraint minimizers. To achieve our goal, we always assume that the minimizers of $I(s,p,\lambda )$ are nonnegative due to functional $F\left(u_{\lambda }\right)\ge F\left(\right|u_{\lambda }\left|\right)$. As follows, we present a detailed analysis of the blow-up rate and location of nonnegative minimizers $u_{\lambda }$ as $\lambda$ tends to ${\lambda }^{\ast }$ from below.

Theorem 3.

Assume that $\left(V_1\right)$ and $\left(V_2\right)$ hold, and $p=\frac{4(s+1)}{3}<4$ for $n=3$, $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$. Let $u_{\lambda }$ denote nonnegative minimizers of $I(s,p,\lambda )$ for any $\lambda <{\lambda }^{\ast }$, then there exists a $x_{\lambda }\in {\mathbb{R}}^n$ such that, as $\lambda \nearrow {\lambda }^{\ast }$

$${ϵ}_{\lambda }^{\frac{n}{2}}{u}_{\lambda }({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })\to \frac{w_p\left(\right|x\left|\right)}{\parallel w_p{\parallel }_2}\mathit{strongly}\mathit{in}{H}^1\left({\mathbb{R}}^n\right),$$

where $w_p$ is given by (5) and ${ϵ}_{\lambda }:={\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx\right)}^{-\frac{1}{2}}$. Furthermore, ${ϵ}_{\lambda }$ and $x_{\lambda }$ satisfy

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}\frac{{ϵ}_{\lambda }}{{\left({\lambda }^{\ast }-\lambda \right)}^{\frac{1}{2(s+1)+q}}}={\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{1}{2(s+1)+q}}$$

and

$${ϵ}_{\lambda }{x}_{\lambda }\to x_i\mathrm{as}\lambda \nearrow {\lambda }^{\ast },$$

where $x_i\in \mathcal{Z}$, and $\mathcal{Z}$ denotes the set of flattest global minima of $V\left(x\right)$.

We remark that the blow-up behavior of nonnegative minimizers presented in Theorem 3 is quite different from the results in [18,20,21,22,24], due to our minimization problem (2) containing a varying nonlocal term. From Theorem 3, one may also observe that the mass of $u_{\lambda }$ concentrates at (i.e., blow-up) some point $x_i$, where $x_i$ is the flattest global minimum of the potential $V\left(x\right)$. Furthermore, the blow-up rate of $u_{\lambda }$ approximates ${\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{1}{2(s+1)+q}}{({\lambda }^{\ast }-\lambda )}^{\frac{1}{2(s+1)+q}}$ as $\lambda \nearrow {\lambda }^{\ast }$.

The structure of the article is arranged as follows. Section 2 involves the existence and non-existence proof of constraint minimizers for $I(s,p,\lambda )$ when $s,p,\lambda$ fulfill $\left(c_1\right)$–$\left(c_4\right)$. If $p=\frac{4(s+1)}{3}<4$ for $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, in Section 3, we establish the refined lower and upper energy estimations of $I(s,p,{\lambda }_k)$ for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$. A detailed proof of Theorems 2 and 3 is presented in Section 3.

2. Existence and Nonexistence Analyses of Minimizers

In this section, we shall present the proof of Theorem 1 on the existence and nonexistence of constraint minimizers, divided into the following two cases:

Case 1.

If either $\left(c_1\right)$ or $\left(c_4\right)$ holds, then (2) has at least one minimizer.

Proof. 

On the one hand, if $\left(c_1\right)$ holds, for any $u\in \mathcal{S}$, we deduce from (8) that

$$\begin{array}{cc}\hfill F\left(u\right)& \ge \frac{1}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u^{2}dx-\frac{\lambda }{\parallel w_p{\parallel }_{L^2}^p}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{\frac{np}{4}},\hfill \end{array}$$

(15)

where $w_q\left(x\right)$ is given by (5). Because we have $\frac{np}{4}<s+1$ by using $\left(c_1\right)$, it follows that for any sequence $\left\{u_n\right\}\subseteq \mathcal{S}$, the $F\left(u_n\right)$ is bounded uniformly from below.

On the other hand, if $\left(c_4\right)$ holds, then it derives from (8) that for any $u\in \mathcal{S}$, we have the following:

$$\begin{array}{cc}\hfill F\left(u\right)& \ge \frac{1}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u^{2}dx-\frac{\lambda }{\parallel w_p{\parallel }_{L^2}^p}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & =\left(\frac{1}{s+1}-\frac{\lambda }{\parallel w_p{\parallel }_{L^2}^p}\right){\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u^{2}dx\hfill \\ \hfill & =\frac{1}{s+1}\left(\frac{{\lambda }^{\ast }-\lambda }{{\lambda }^{\ast }}\right){\left({\int }_{{\mathbb{R}}^n}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u^{2}dx\ge 0,\hfill \end{array}$$

(16)

where ${\lambda }^{\ast }=\frac{\parallel w_p{\parallel }_{L^2}^p}{s+1}$. Inequality (16) also yields that $F\left(u_n\right)$ is bounded uniformly from below for any sequence $\left\{u_n\right\}\subseteq \mathcal{S}$.

Since $F\left(u_n\right)$ is bounded uniformly from below for any sequence $\left\{u_n\right\}\subseteq \mathcal{S}$, there is a minimization sequence $\left\{u_n\right\}\subseteq \mathcal{S}$ fulfilling

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

(17)

Further, it is not difficult to deduce from (15) and (16) that ${\int }_{{\mathbb{R}}^n}{|\nabla u_n|}^{2}dx$ and ${\int }_{{\mathbb{R}}^n}V\left(x\right){\left|u_n\right|}^{2}dx$ are bounded uniformly for n; that is, $\left\{u_n\right\}$ is bounded in $\mathcal{H}$. By applying the compact embedding Theorem 2.1 in [38], one can see that there exists a $u_0\in \mathcal{S}$, and $\left\{u_n\right\}$ has a subsequence $\left\{u_k\right\}$, such that, as $k\to \infty$, we have the following:

$$u_k\rightharpoonup u_0\mathrm{weakly}\mathrm{in}\mathcal{H},u_k\to u_0\mathrm{strongly}\mathrm{in}{L}^q\left({\mathbb{R}}^n\right),2\le q<2^{\ast }.$$

(18)

Moreover, by applying the weak lower semi-continuity, we have the following:

$$\underset{k\to \infty }{lim\; inf}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u_k|}^{2}dx\right)}^{s+1}\ge {\left({\int }_{{\mathbb{R}}^n}{|\nabla u_0|}^{2}dx\right)}^{s+1}$$

which, together with (18), yields the following:

$$I(s,p,\lambda )=\underset{k\to \infty }{lim\; inf}F\left(u_k\right)\ge F\left(u_0\right)\ge I(s,p,\lambda ).$$

The above statements show that $F\left(u_0\right)=I(s,p,\lambda )$, hence $u_0$ is a minimizer of $I(s,p,\lambda )$. □

Case 2.

If either $\left(c_2\right)$ or $\left(c_3\right)$ holds, then (2) has no minimizer.

Proof. 

The nonexistence proof of the minimizer becomes true by estimating $I(s,p,\lambda )=-\infty$ or $I(s,p,{\lambda }^{\ast })=0$. To obtain this, a suitable test function is established, such as the following:

$$u_{\tau }\left(x\right):=\frac{{\mathcal{A}}_{\tau }{\tau }^{\frac{n}{2}}}{\parallel w_p{\parallel }_2}\phi (x-\widehat{x})w_p\left(\tau \right|x-\widehat{x}\left|\right),$$

(19)

where $w_p\left(x\right)$ is given by (5), $\widehat{x}$ satisfies $V\left(\widehat{x}\right)=0$, with $\tau >0$ being a real number. The above $\phi \left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^n\right)$ is selected, such that $0\le \phi \left(x\right)\le 1$, $\phi \left(x\right)=1$ for $\left|x\right|\le 1$, $\phi \left(x\right)=0$ for $\left|x\right|\ge 2$, and $|\nabla \phi (x\left)\right|\le 2$ for any $x\in {\mathbb{R}}^n$. Note that ${\mathcal{A}}_{\tau }$ in (19) is chosen so that $\parallel u_{\tau }{\parallel }_{L^2\left({\mathbb{R}}^n\right)}^2=1$. Under the assumptions of $\left(c_2\right)$ and $\left(c_3\right)$, we next establish some estimations of ${\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\tau }|}^{2}dx\right)}^{s+1}$, ${\int }_{{\mathbb{R}}^n}V\left(x\right)u_{\tau }^{2}dx$ and ${\int }_{{\mathbb{R}}^n}{\left|u_{\tau }\right|}^{p+2}dx$, which arise in (3).

Since $\parallel u_{\tau }{\parallel }_{L^2}^2=1$, one obtains from (7) that ${\mathcal{A}}_{\tau }$ in (20) satisfies the following:

$$1\le A_{\tau }\le 1+g\left(\tau \right)\mathrm{and}\underset{\tau \to \infty }{lim}{A}_{\tau }=1,$$

(20)

where $g\left(\tau \right)=O\left({\tau }^{-\infty }\right)$ is a function satisfying the following: $\underset{\tau \to \infty }{lim}\left|g\left(\tau \right)\right|{\tau }^s=0$ for any $s>0$. Applying the exponential decay property (7) and (20), there exist a ball $B_{\tau }\left(0\right)\subseteq {\mathbb{R}}^n$ and constant $C>0$ satisfying the following:

$$\begin{array}{cc}\hfill {\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\tau }|}^{2}dx\right)}^{s+1}& =\frac{A_{\tau }^{2(s+1)}{\tau }^{2(s+1)}}{\parallel w_p{\parallel }_2^{2(s+1)}}[{\left({\int }_{{\mathbb{R}}^n\setminus B_{\tau }\left(0\right)}|\nabla (\phi (x-\widehat{x})w_p\left(\tau \right|x-\widehat{x}{\left|\right)\left)\right|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +{\left({\int }_{B_{\tau }\left(0\right)}|\nabla (\phi (x-\widehat{x})w_p\left(\tau \right|x-\widehat{x}{\left|\right)\left)\right|}^{2}dx\right)}^{s+1}]\hfill \\ \hfill & \le \frac{A_{\tau }^{2(s+1)}{\tau }^{2(s+1)}}{\parallel w_p{\parallel }_2^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^n}{|\nabla w_p|}^{2}dx\right)}^{s+1}+C{e}^{-2(s+1)\tau }\hfill \\ \hfill & \le \frac{{\tau }^{2(s+1)}}{\parallel w_p{\parallel }_2^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^n}{|\nabla w_p|}^{2}dx\right)}^{s+1}+C{e}^{-2(s+1)\tau }+O\left({\tau }^{-\infty }\right)\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^n}{\left|u_{\tau }\right|}^{p+2}dx& =\frac{A_{\tau }^{p+2}{\tau }^{\frac{pn}{2}}}{\parallel w_p{\parallel }_2^{p+2}}{\int }_{{\mathbb{R}}^n\setminus B_{\tau }\left(0\right)}|\phi (x-\widehat{x})w_p\left(\tau \right|x-\widehat{x}{\left|\right)|}^{p+2}dx\hfill \\ \hfill & +\frac{A_{\tau }^{p+2}{\tau }^{\frac{pn}{2}}}{\parallel w_p{\parallel }_2^{p+2}}{\int }_{B_{\tau }\left(0\right)}|\phi (x-\widehat{x})w_p\left(\tau \right|x-\widehat{x}{\left|\right)|}^{p+2}dx\hfill \\ \hfill & \ge \frac{A_{\tau }^{p+2}{\tau }^{\frac{pn}{2}}}{\parallel w_p{\parallel }_2^{p+2}}{\int }_{{\mathbb{R}}^n}{\left|w_p\right|}^{p+2}dx-C{e}^{-(1+p)\tau }\hfill \\ \hfill & \ge \frac{{\tau }^{\frac{pn}{2}}}{\parallel w_p{\parallel }_2^{p+2}}{\int }_{{\mathbb{R}}^n}{\left|w_p\right|}^{p+2}dx-C{e}^{-(1+p)\tau }+O\left({\tau }^{-\infty }\right).\hfill \end{array}$$

(22)

Since $\left(V_1\right)$ holds, one can deduce that there is a ball $B_{\sqrt{\tau }}\left(0\right)\subseteq {\mathbb{R}}^n,$ such that, as $\tau \to \infty$, we have the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^n}V\left(x\right){\left|u_{\tau }\right|}^{2}dx& =\frac{A_{\tau }^2}{\parallel w_p{\parallel }_2^2}{\int }_{{\mathbb{R}}^n\setminus B_{\sqrt{\tau }}\left(0\right)}V(\frac{x}{\tau }+\widehat{x}){\phi }^2\left(\frac{x}{\tau }\right){\left|w_p\left(x\right)\right|}^{2}dx\hfill \\ \hfill & +\frac{A_{\tau }^2}{\parallel w_p{\parallel }_2^2}{\int }_{B_{\sqrt{\tau }}\left(0\right)}V(\frac{x}{\tau }+\widehat{x}){\phi }^2\left(\frac{x}{\tau }\right){\left|w_p\left(x\right)\right|}^{2}dx\hfill \\ \hfill & \le \frac{A_{\tau }^2}{\parallel w_p{\parallel }_2^2}{\int }_{B_{\sqrt{\tau }}\left(0\right)}V(\frac{x}{\tau }+\widehat{x}){\left|w_p\left(x\right)\right|}^{2}dx+C{e}^{-\sqrt{\tau }}\hfill \\ \hfill & \le V\left(\widehat{x}\right)+C{e}^{-\sqrt{\tau }}+O\left({\tau }^{-\infty }\right)\hfill \end{array}$$

(23)

which together with (21) and (22), implies that, as $\tau \to \infty$, we have the following:

$$\begin{array}{cc}\hfill I(s,p,\lambda )\le F\left(u_{\tau }\right)& =\frac{A_{\tau }^{2(s+1)}{\tau }^{2(s+1)}}{(s+1)\parallel w_p{\parallel }_2^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^n}{|\nabla w_p|}^{2}dx\right)}^{s+1}+V\left(\widehat{x}\right)\hfill \\ \hfill & -\frac{2\lambda A_{\tau }^{p+2}{\tau }^{\frac{pn}{2}}}{(p+2)\parallel w_p{\parallel }_2^{p+2}}{\int }_{{\mathbb{R}}^n}{\left|w_p\right|}^{p+2}dx+C{e}^{-\sqrt{\tau }}+O\left({\tau }^{-\infty }\right).\hfill \end{array}$$

(24)

If ($c_2$) holds, one has $\frac{pn}{2}>2(s+1)$ which yields from (24) that $I(s,p,\lambda )\le F\left(u_{\tau }\right)\to -\infty$ as $\tau \to \infty$. Hence, $I(s,p,\lambda )$ has no minimizer.

If $(c_3$) holds, we first consider the case of $\lambda >{\lambda }^{\ast }$. Combining (6), (20), and (24), one obtains the following:

$$\begin{array}{cc}\hfill I(s,p,\lambda )\le F\left(u_{\tau }\right)& =\frac{A_{\tau }^{2(s+1)}{\tau }^{2(s+1)}}{(s+1)\parallel w_p{\parallel }_2^{2(s+1)}}{\left({\int }_{{\mathbb{R}}^n}{\left|w_p\right|}^{2}dx\right)}^{s+1}+V\left(\widehat{x}\right)\hfill \\ \hfill & -\frac{2\lambda A_{\tau }^{p+2}{\tau }^{\frac{pn}{2}}}{(p+2)\parallel w_p{\parallel }_2^{p+2}}\frac{p+2}{2}{\int }_{{\mathbb{R}}^n}{\left|w_p\right|}^{2}dx+o\left(1\right)+O\left({\tau }^{-\infty }\right)\hfill \\ \hfill & =\frac{{\tau }^{2(s+1)}}{s+1}-\frac{\lambda {\tau }^{\frac{pn}{2}}}{\parallel w_p{\parallel }_2^p}+o\left(1\right)+O\left({\tau }^{-\infty }\right)\hfill \\ \hfill & =\frac{{\tau }^{2(s+1)}}{s+1}\left(\frac{{\lambda }^{\ast }-\lambda }{\lambda }\right)+o\left(1\right)+O\left({\tau }^{-\infty }\right)\hfill \end{array}$$

(25)

which also yields that $I(s,p,\lambda )=-\infty$ due to $\lambda >{\lambda }^{\ast }$. It, thus, follows that $I(s,p,\lambda )$ has no minimizer when $(c_3$) holds.

For the other case of $\lambda ={\lambda }^{\ast }$, one can conclude from (16) and (25) that $I(s,p,{\lambda }^{\ast })=0$. In view of this fact, we next show that $I(s,p,{\lambda }^{\ast })$ has no minimizer. Assume, toward a contradiction that $I(s,p,{\lambda }^{\ast })$ has a nontrivial minimizer $u_0\in \mathcal{S}$, and as pointed out in Section 1, we always assume that $u_0$ is nonnegative. Because $V\left(x\right)\ge 0$ and the unique optimizer of the Gagliardo–Nirenberg inequality (8) is attained for $u=w_p$, one then obtains from (16) that $w_p={\parallel w_p\parallel }_{L^2}{u}_0$ (up to the translations). Furthermore, $u_0$ satisfies the following:

$${\int }_{{\mathbb{R}}^n}V\left(x\right)u_0^{2}dx=\underset{{\mathbb{R}}^n}{min}V\left(x\right)=0$$

and

$$\frac{1}{(s+1)}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u_0|}^{2}dx\right)}^{s+1}=\frac{2{\lambda }^{\ast }}{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_0\right|}^{p+2}dx.$$

However, this is a contradiction because the above two equalities cannot hold at the same time. In fact, the first equality holds for $u_0$ having compact support, while the second one implies that $u_0$ has no compact support. Therefore, $I(s,p,{\lambda }^{\ast })$ has no minimizer.

At last, we shall prove that $\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}I(s,p,\lambda )=0$. Using the test function defined by (19) and taking

$$\tau ={({\lambda }^{\ast }-\lambda )}^{-\frac{1}{4(s+1)}}\mathrm{as}\lambda \nearrow {\lambda }^{\ast },$$

one then deduces from (25) that

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; sup}I(s,p,\lambda )\le F\left(u_{\tau }\right)=\frac{{({\lambda }^{\ast }-\lambda )}^{\frac{1}{2}}}{\lambda (s+1)}+o\left(1\right)+O\left({\tau }^{-\infty }\right)\to 0.$$

(26)

By (16), we can obtain $\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}I(s,p,\lambda )\ge 0$, which together with (26) gives $\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}I(s,p,\lambda )=I(s,p,{\lambda }^{\ast })=0$. This completes the proof of Theorem 1. □

3. Refined Energy Estimation

This section is mainly concerned with the refined energy estimation of $I(s,p,{\lambda }_k)$ for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$. We begin with the upper energy estimation of $I(s,p,\lambda )$, which is described in the following lemma.

Lemma 1.

Under the assumptions of $\left(V_1\right)$, $\left(V_2\right)$, $p=\frac{4(s+1)}{3}<4$ for $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, then for any λ with $\lambda \nearrow {\lambda }^{\ast }$, $I(s,p,\lambda )$ satisfies the following:

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; sup}\frac{I(s,p,\lambda )}{{\left({\lambda }^{\ast }-\lambda \right)}^{\frac{q}{2(s+1)+q}}}\le \frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}}+o\left(1\right),$$

(27)

where $q,\alpha$ are given by (10) and (12).

Proof. 

Choosing the same test function defined by (19) and $\widehat{x}$ replaced by some $x_i\in \mathcal{Z}$, then it can be deduced from (7), (9)–(13), and (19) that there is a ball $B_{\sqrt{\tau }}\left(0\right)\subseteq {\mathbb{R}}^n$, such that, as $\tau \to \infty$, we have the following:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^n}V\left(x\right)u_{\tau }^{2}dx=& \frac{A_{\tau }^2}{\parallel w_p{\parallel }_{L^2}^2}{\int }_{{\mathbb{R}}^n}V(\frac{x}{\tau }+x_i){\phi }^2\left(\frac{x}{\tau }\right){\left|w_p\right|}^{2}dx\hfill \\ \hfill \le & \frac{A_{\tau }^2}{\parallel w_p{\parallel }_{L^2}^2}{\int }_{B_{\sqrt{\tau }}\left(0\right)}V(\frac{x}{\tau }+x_i){\left|w_p\right|}^{2}dx+O\left({\tau }^{-\infty }\right)\hfill \\ \hfill =& \frac{1}{\parallel w_p{\parallel }_{L^2}^2}{\int }_{B_{\sqrt{\tau }}\left(0\right)}h(\frac{x}{\tau }+x_i)\prod _{j=1}^n|\frac{x}{\tau }+x_i-x_j{|}^{q_j}{\left|w_p\right|}^{2}dx+O\left({\tau }^{-\infty }\right)\hfill \\ \hfill =& {\tau }^{-q}\frac{1}{\parallel w_p{\parallel }_{L^2}^2}\underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^2}{\left|x\right|}^q{\left|w_p\left(x\right)\right|}^{2}dx+o\left({\tau }^{-q}\right)\hfill \\ \hfill =& \alpha {\tau }^{-q}+o\left({\tau }^{-q}\right),\hfill \end{array}.$$

(28)

Furthermore, similar to the estimations of (21) and (22), we have the following:

$$\frac{1}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\tau }|}^{2}dx\right)}^{s+1}-\frac{2\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_{\tau }\right|}^{p+2}dx\le \frac{{\tau }^{2(s+1)}}{s+1}\left(\frac{{\lambda }^{\ast }-\lambda }{\lambda }\right)+O\left({\tau }^{-\infty }\right)$$

(29)

which together with (28) yields

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; sup}I(s,p,\lambda )\le \frac{{\tau }^{2(s+1)}}{\lambda (s+1)}({\lambda }^{\ast }-\lambda )+\alpha {\tau }^{-q}+o\left({\tau }^{-q}\right).$$

(30)

Taking $\tau ={\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{1}{2(s+1)+q}}{\left({\lambda }^{\ast }-\lambda \right)}^{-\frac{1}{2(s+1)+q}}$, one then derives the following from (30):

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; sup}\frac{I(s,p,\lambda )}{{\left({\lambda }^{\ast }-\lambda \right)}^{\frac{q}{2(s+1)+q}}}\le \frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}}+o\left(1\right)$$

(31)

which gives (27). □

In order to estimate the lower bound of $I(s,p,\lambda )$, we assume that $u_{\lambda }$ is a nonnegative minimizer of (2), and $u_{\lambda }$ fulfills the following elliptic equation:

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

(32)

where ${\mu }_{\lambda }\in \mathbb{R}$ is a suitable Lagrange multiplier. We define a function, as follows:

$${\widehat{v}}_{\lambda }\left(x\right):={ϵ}_{\lambda }^{\frac{n}{2}}{u}_{\lambda }\left({ϵ}_{\lambda }x\right)$$

(33)

and ${ϵ}_{\lambda }$ is given by the following:

$${ϵ}_{\lambda }:={\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx\right)}^{-\frac{1}{2}}>0.$$

(34)

In the following, some indispensable conclusions are established for estimating lower energy and are divided into Propositions 1–4, as follows:

Proposition 1.

${ϵ}_{\lambda }$ and ${\widehat{v}}_{\lambda }$ defined by (33) and (34) satisfy

$${ϵ}_{\lambda }\to 0\mathrm{as}\lambda \nearrow {\lambda }^{\ast }$$

and

$${\int }_{{\mathbb{R}}^n}|\nabla {\widehat{v}}_{\lambda }{|}^{2}dx=1\mathrm{and}{\int }_{{\mathbb{R}}^n}{\left|{\widehat{v}}_{\lambda }\right|}^{p+2}\to \frac{p+2}{2{\lambda }^{\ast }(s+1)}\mathrm{as}\lambda \nearrow {\lambda }^{\ast }.$$

In fact, if ${ϵ}_{\lambda }\nrightarrow 0$, then we can select a sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$ such that $\left\{u_{{\lambda }_k}\right\}$ is bounded uniformly in $\mathcal{H}$. We repeat the proof procedure of Case 1 in Theorem 1 and conclude that $I(s,p,{\lambda }^{\ast })$ has at least one minimizer. However, Theorem 1 implies that $I(s,p,{\lambda }^{\ast })$ has no minimizer. Thus, ${ϵ}_{\lambda }\to 0$ as $\lambda \nearrow {\lambda }^{\ast }$ holds. By definitions (33) and (34), we have that

$${\int }_{{\mathbb{R}}^n}|\nabla {\widehat{v}}_{\lambda }{|}^{2}dx={ϵ}_{\lambda }^{-2}{\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx=1.$$

Further, using the conclusions in Theorem 1, as $\lambda \nearrow {\lambda }^{\ast }$, we deduce the following:

$$\begin{array}{cc}\hfill 0& \le \frac{1}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx\right)}^{s+1}-\frac{2\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_{\lambda }\right|}^{p+2}dx\hfill \\ \hfill & =\frac{1}{s+1}{ϵ}_{\lambda }^{-2(s+1)}-\frac{2\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_{\lambda }\right|}^{p+2}dx\hfill \\ \hfill & =\frac{1}{s+1}{ϵ}_{\lambda }^{-2(s+1)}-\frac{2\lambda }{p+2}{ϵ}_{\lambda }^{-\frac{pn}{2}}{\int }_{{\mathbb{R}}^n}{\left|{\widehat{v}}_{\lambda }\right|}^{p+2}dx\hfill \\ \hfill & \le I(s,p,\lambda )\to 0.\hfill \end{array}$$

(35)

Since $p=\frac{4(s+1)}{3}<4$ for $n=3$ and $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, (33) together with (35) yields the following: $\lambda \nearrow {\lambda }^{\ast }$

$$\frac{1}{s+1}-\frac{2\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|{\widehat{v}}_{\lambda }\right|}^{p+2}dx\to 0,$$

that is,

$${\int }_{{\mathbb{R}}^n}{\left|{\widehat{v}}_{\lambda }\right|}^{p+2}dx\to \frac{p+2}{2{\lambda }^{\ast }(s+1)}.$$

Thus, we complete the proof of Proposition 1.

Proposition 2.

There exist $x_{\lambda }\subset {\mathbb{R}}^n$ and positive constants $s,\mathcal{M}$ fulfilling the following:

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}{\int }_{B_s\left(x_{\lambda }\right)}{\left|{\widehat{v}}_{\lambda }\left(x\right)\right|}^{2}dx\ge \mathcal{M}>0.$$

(36)

If (36) is not true, then for any $s>0$ and $y\in {\mathbb{R}}^n$, there admits a sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$ such that ${\widehat{v}}_{{\lambda }_k}$ satisfies the following:

$$\underset{k\to \infty }{lim\; sup}{\int }_{B_s\left(y\right)}{\left|{\widehat{v}}_{{\lambda }_k}\right|}^{2}dx=0.$$

Applying the vanishing Lemma 1.1 in [39], one knows that ${\widehat{v}}_{{\lambda }_k}\to 0$ in $L^{\nu }\left({\mathbb{R}}^n\right)$ for all $\nu \in (2,2^{\ast })$. In particular, we have

$${\int }_{{\mathbb{R}}^n}{\left|{\widehat{v}}_{{\lambda }_k}\right|}^{p+2}dx\to 0\mathrm{as}k\to \infty$$

which is a contradiction with Proposition 1. Therefore, Proposition 2 holds.

Set

$$v_{\lambda }\left(x\right):={ϵ}_{\lambda }^{\frac{n}{2}}{u}_{\lambda }({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda }),$$

(37)

where $x_{\lambda }$ is given by (36). By Propositions 1 and 2, we know that there exist positive constants $s,\mathcal{M}$ satisfying

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}{\int }_{B_s\left(0\right)}{\left|v_{\lambda }\left(x\right)\right|}^{2}dx\ge \mathcal{M}>0.$$

(38)

Further, $v_{\lambda }$ fulfills the following:

$${\int }_{{\mathbb{R}}^n}|\nabla v_{\lambda }{|}^{2}dx=1\mathrm{and}{\int }_{{\mathbb{R}}^n}{\left|v_{\lambda }\right|}^{p+2}\to \frac{p+2}{2{\lambda }^{\ast }(s+1)}\mathrm{as}\lambda \nearrow {\lambda }^{\ast }.$$

(39)

Proposition 3.

For any sequence $\left\{{\lambda }_k\right\}$ as ${\lambda }_k\nearrow {\lambda }^{\ast }$ with $k\to \infty$, there exists $y_0\in {\mathbb{R}}^n$, such that $v_{{\lambda }_k}$ defined by (37) satisfies the following:

$$\underset{k\to \infty }{lim}{v}_{{\lambda }_k}=\frac{w_p\left(\right|x-y_0\left|\right)}{\parallel w_p{\parallel }_{L^2}},$$

(40)

strongly in $H^1\left({\mathbb{R}}^n\right)$ and $w_p\left(x\right)$ given by (5).

Because $u_{\lambda }$ is the nonnegative minimizer of (2), it follows from (32) and (39) that for $p=\frac{4(s+1)}{n}$, $v_{\lambda }\left(x\right)$ satisfies the following:

$$-\Delta v_{\lambda }+{ϵ}_{\lambda }^{2(s+1)}V({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })v_{\lambda }={\mu }_{\lambda }{ϵ}_{\lambda }^{2(s+1)}{v}_{\lambda }+\lambda {\left|v_{\lambda }\right|}^p{v}_{\lambda },x\in {\mathbb{R}}^n.$$

(41)

Theorem 1 shows that $\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}I(s,p,\lambda )\to 0$, which gives

$${ϵ}_{\lambda }^{2(s+1)}I(s,p,\lambda )\to 0\mathrm{as}\lambda \nearrow {\lambda }^{\ast }.$$

(42)

By applying (3) and (32), ${\mu }_{\lambda }$ fulfills the following:

$$\begin{array}{cc}\hfill {\mu }_{\lambda }& =I(s,p,\lambda )+\frac{s}{s+1}{\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{\lambda }|}^{2}dx\right)}^{s+1}-\frac{p\lambda }{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_{\lambda }\right|}^{p+2}dx,\hfill \end{array}$$

(43)

which, together with Proposition 1 and (42), gives the following: $\lambda \nearrow {\lambda }^{\ast }$

$${\mu }_{\lambda }{ϵ}_{\beta }^{2(s+1)}\to -\frac{p-2s}{2(s+1)}\mathrm{and}{\epsilon }_{\lambda }^{2(s+1)}{\int }_{{\mathbb{R}}^n}V({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })v_{\lambda }^2\left(x\right)dx\to 0,$$

(44)

where $\frac{p-2s}{2(s+1)}=\frac{1}{s+1}-1+\frac{2}{n}>0$.

Using Proposition 1, (41), and (44), for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$ as $k\to \infty$, and passing to a subsequence if necessary, we have $v_{{\lambda }_k}\rightharpoonup v_0$ in $H\left({\mathbb{R}}^n\right)$ for some $v_0\in H^1\left({\mathbb{R}}^n\right)$. Passing the weak limit to (41), it follows that $v_0$ fulfills the following:

$$-\Delta v_0+\left[\frac{2}{n}-1+\frac{1}{s+1}\right]v_0={\lambda }^{\ast }{\left|v_0\right|}^p{v}_0,x\in {\mathbb{R}}^n.$$

(45)

Since $p=\frac{4(s+1)}{n}$, the elliptic Equation (5) is easily converted into the following:

$$-\Delta w+\left[\frac{2}{n}-1+\frac{1}{s+1}\right]w=\frac{1}{s+1}{\left|w\right|}^{p}w,x\in {\mathbb{R}}^n,n\le 3,0<p<2^{\ast }-2,$$

(46)

where (46) admits a unique (up to translations) positive radially symmetric solution $w_p\in H^1\left({\mathbb{R}}^n\right)$. Since ${\lambda }^{\ast }=\frac{1}{s+1}{\parallel w_p\parallel }_{L^2}^p$, a simple analysis yields that $v_0$ satisfies

$$v_0\left(x\right)=\frac{w_p\left(\right|x-y_0\left|\right)}{\parallel w_p{\parallel }_{L^2}}\mathit{for}\mathit{some}{y}_0\in {\mathbb{R}}^n$$

(47)

and

$$\underset{k\to \infty }{lim}{\int }_{{\mathbb{R}}^n}|v_{{\lambda }_k}{|}^{2}dx={\int }_{{\mathbb{R}}^n}{\left|v_0\right|}^{2}dx=1,$$

(48)

which shows that $v_{{\lambda }_k}\to v_0$ strongly in $L^2\left({\mathbb{R}}^n\right)$ as $k\to \infty$. Making full use of Hölder and Sobolev inequalities, one also obtains that ${\parallel u\parallel }_{L^q\left({\mathbb{R}}^n\right)}\le {C\parallel u\parallel }_{L^2\left({\mathbb{R}}^n\right)}^{\gamma }{\parallel u\parallel }_{H^1\left({\mathbb{R}}^n\right)}^{1-\gamma }$ for any $u\in H^1\left({\mathbb{R}}^n\right)$ with $q\in (2,2^{\ast })$ and $\gamma \in (0,1)$. We then conclude that $v_{{\lambda }_k}\to v_0$ strongly in $L^q\left({\mathbb{R}}^n\right)$ with $q\in [2,2^{\ast })$ as $k\to \infty$. It, thus, follows from (41), (44), and (48) that $\underset{k\to \infty }{lim}\parallel \nabla v_{{\lambda }_k}{\parallel }_2^2={\parallel \nabla v_0\parallel }_2^2$. This completes the proof of Proposition 3.

Proposition 4.

The ${ϵ}_{\lambda }$ and $x_{\lambda }$ given by Propositions 1 and 2 fulfill that ${ϵ}_{\lambda }{x}_{\lambda }$ is bounded uniformly as $\lambda \nearrow {\lambda }^{\ast }$, and for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, passing to a subsequence if necessary (still denoted by $\left\{{\lambda }_k\right\}$), the ${ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}$ satisfies the following:

$${ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}\to x_i\mathit{and}\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_i|}{{ϵ}_{{\lambda }_k}}\mathit{is}\mathit{bounded}\mathit{uniformly}\mathit{as}k\to \infty ,$$

(49)

where $x_i\in \mathcal{Z}$; that is, $x_i$ is one of the flattest global minima of $V\left(x\right)$

Toward a contradiction, suppose that ${ϵ}_{\lambda }{x}_{\lambda }$ is unbounded uniformly as $\lambda \nearrow {\lambda }^{\ast }$; that is, ${ϵ}_{\lambda }{x}_{\lambda }\to \infty$ as $\lambda \nearrow {\lambda }^{\ast }$. In truth, Theorem 1 implies that $\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}I(s,p,\lambda )=0$, which together with (8) gives the following:

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}{\int }_{{\mathbb{R}}^n}V\left(x\right)u_{\lambda }^{2}dx=\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}{\int }_{{\mathbb{R}}^n}V({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })v_{\lambda }^{2}dx\to 0.$$

(50)

However, by applying (38) and Fatou’s Lemma, there exists a constant $C>0$, such that

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}{\int }_{{\mathbb{R}}^n}V({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })v_{\lambda }^{2}dx\ge {\int }_{{\mathbb{R}}^n}\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}V({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })v_{\lambda }^{2}dx\ge C\mathcal{M}>0$$

(51)

which contradicts (50). Therefore, ${ϵ}_{\lambda }{x}_{\lambda }$ is bounded uniformly as $\lambda \nearrow {\lambda }^{\ast }$. This then yields that for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, passing to a subsequence if necessary (still denoted by $\left\{{\lambda }_k\right\}$), there exists a $x_0\in {\mathbb{R}}^n$, such that ${ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}$ satisfies the following:

$${ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}\to x_0\mathit{as}{\lambda }_k\nearrow {\lambda }^{\ast }.$$

(52)

Moreover, $x_0$ is a minimum value of $V\left(x\right)$ fulfilling $V\left(x_0\right)=0$. If not, and $V\left(x_0\right)>0$, we still obtain a contradiction by repeating the proof of (51).

We next claim that the following, i.e.,

$$\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_0|}{{ϵ}_{{\lambda }_k}}\mathit{is}\mathit{bounded}\mathit{uniformly}\mathit{as}{\lambda }_k\nearrow {\lambda }^{\ast }.$$

(53)

If (53) is not true, we claim that the following, i.e., $\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_0|}{{ϵ}_{{\lambda }_k}}\to \infty$ as $k\to \infty$, then one derives from $\left(V_2\right)$ and (38) that for any large M

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

(54)

Similar to the estimation of (16), we have the following:

$$\begin{array}{c}\hfill I(s,p,{\lambda }_k)\ge \frac{1}{s+1}\left(\frac{{\lambda }^{\ast }-\lambda }{{\lambda }^{\ast }}\right){ϵ}_{{\lambda }_k}^{-2(s+1)}+M{ϵ}_{{\lambda }_k}^{q_{i_0}}\ge \mathcal{C}{\left({\lambda }^{\ast }-{\lambda }_k\right)}^{\frac{q}{2(s+1)+q}}\mathrm{as}{\lambda }_k\nearrow {\lambda }^{\ast },\end{array}$$

where $\mathcal{C}$ is an arbitrarily large constant, which contradicts the energy upper bound in Lemma 1. Thus, (53) holds.

Finally, we shall prove that there exists some $x_i\in \mathcal{Z}$ such that $x_0=x_i$. If this is false, by considering (53) and repeating the proof of (54), one has the following:

$$\underset{k\to \infty }{lim\; inf}\frac{1}{{ϵ}_{{\lambda }_k}^{p_{i_0}}}{\int }_{{\mathbb{R}}^n}V({ϵ}_{{\lambda }_k}x+x_{{\lambda }_k}{ϵ}_{{\lambda }_k})v_{{\lambda }_k}^{2}dx\ge C_0\left(R\right)>0.$$

(55)

From this, one can derive that ${\lambda }_k\nearrow {\lambda }^{\ast }$, as follows:

$$\begin{array}{c}\hfill I(s,p,{\lambda }_k)\ge \frac{1}{s+1}\left(\frac{{\lambda }^{\ast }-\lambda }{{\lambda }^{\ast }}\right){ϵ}_{{\lambda }_k}^{-2(s+1)}+C_0\left(R\right){ϵ}_{{\lambda }_k}^{p_{i_0}}\ge \mathcal{D}{\left({\lambda }^{\ast }-{\lambda }_k\right)}^{\frac{q_{i_0}}{2(s+1)+q_{i_0}}}\end{array}$$

where $\mathcal{D}>0$ is constant and $q_{i_0}>q$, which also contradicts Lemma 1. We repeat the proof of (53); one can obtain that $\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_i|}{{ϵ}_{{\lambda }_k}}$ is bounded uniformly as ${\lambda }_k\nearrow {\lambda }^{\ast }$. We, thus, have completed the proof of Proposition 4.

Based on the above Proposition 1–Proposition 4, we next establish the estimation of the energy lower bound for $I(s,p,{\lambda }_k)$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, as follows:

Lemma 2.

If $\left(V_1\right)$, $\left(V_2\right)$ holds and $p=\frac{4(s+1)}{3}<4$ for $n=3$, $p=\frac{4(s+1)}{n}<\infty$ for $n=1,2$, then for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, passing a subsequence if necessary (still denoted by $\left\{{\lambda }_k\right\}$), $I(s,p,{\lambda }_k)$ satisfies the following:

$$\underset{{\lambda }_k\nearrow {\lambda }^{\ast }}{lim\; inf}\frac{I(s,p,{\lambda }_k)}{{\left({\lambda }^{\ast }-{\lambda }_k\right)}^{\frac{q}{2(s+1)+q}}}\ge \frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}},$$

(56)

where $q,\alpha$ are positive constants defined by (10) and (12).

Proof. 

Assume that $u_{\lambda }$ is a nonnegative minimizer of (2). For any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$, taking a subsequence if necessary (still denoted by $\left\{{\lambda }_k\right\}$), one then derives from (8) and Proposition 1 the following:

$$\begin{array}{cc}\hfill I(s,p,{\lambda }_k)& =F\left(u_{{\lambda }_k}\right)\hfill \\ \hfill & ={\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{{\lambda }_k}|}^{2}dx\right)}^{s+1}-\frac{2{\lambda }_k}{p+2}{\int }_{{\mathbb{R}}^n}{\left|u_{{\lambda }_k}\right|}^{p+2}dx+{\int }_{{\mathbb{R}}^n}V\left(x\right)u_{{\lambda }_k}^{2}dx\hfill \\ \hfill & \ge \frac{1}{s+1}\left(\frac{{\lambda }^{\ast }-{\lambda }_k}{{\lambda }^{\ast }}\right){\left({\int }_{{\mathbb{R}}^n}{|\nabla u_{{\lambda }_k}|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^n}V\left(x\right)u_{{\lambda }_k}^{2}dx\hfill \\ \hfill & =\frac{{\lambda }^{\ast }-{\lambda }_k}{(s+1){\lambda }^{\ast }}{ϵ}_{{\lambda }_k}^{-2(s+1)}+{\int }_{{\mathbb{R}}^n}V({ϵ}_{{\lambda }_k}x+{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k})v_{{\lambda }_k}^{2}dx.\hfill \end{array}$$

(57)

Because Proposition 4 gives that $\left\{\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_i|}{{ϵ}_{{\lambda }_k}}\right\}$ is bounded uniformly as ${\lambda }_k\nearrow {\lambda }^{\ast }$ and $x_i\in \mathcal{Z}$, we have $\frac{|{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k}-x_i|}{{ϵ}_{{\lambda }_k}}\to \overline{x}$ for some $\overline{x}\in {\mathbb{R}}^n$. Combining (9)–(13) and Proposition 3, one then deduces the following:

$$\begin{array}{cc}\hfill {\displaystyle \underset{{\lambda }_k\nearrow {\lambda }^{\ast }}{lim\; inf}}\frac{1}{{ϵ}_{{\lambda }_k}^q}{\int }_{{\mathbb{R}}^n}V({ϵ}_{{\lambda }_k}x+{ϵ}_{{\lambda }_k}{x}_{{\lambda }_k})v_{{\lambda }_k}^{2}dx& =\underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^n}|x+\overline{x}{|}^q{v}_0^{2}dx\hfill \\ & \ge \underset{x\to x_i}{lim}\frac{V\left(x\right)}{|x-x_i{|}^q}{\int }_{{\mathbb{R}}^n}{\left|x\right|}^q{v}_0^{2}dx=\alpha .\hfill \end{array}$$

(58)

Notice that the above equality holds only for $\overline{x}=0$; hence, one calculates the following from (57) to (58), as well as Proposition 1–Proposition 3:

$$\underset{{\lambda }_k\nearrow {\lambda }^{\ast }}{lim\; inf}I(s,p,{\lambda }_k)\ge \frac{{\lambda }^{\ast }-{\lambda }_k}{(s+1){\lambda }^{\ast }}{ϵ}_{{\lambda }_k}^{-2(s+1)}+\alpha {ϵ}_{{\lambda }_k}^q.$$

(59)

Since the energy upper bound is constrained as described in Lemma 1, one can take the infimum over ${ϵ}_{{\lambda }_k}$ and it must satisfy the following:

$$\underset{{\lambda }_k\nearrow {\lambda }^{\ast }}{lim}\frac{{ϵ}_{{\lambda }_k}}{{\left({\lambda }^{\ast }-{\lambda }_k\right)}^{\frac{1}{2(s+1)+q}}}={\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{1}{2(s+1)+q}},$$

(60)

which also yields the following:

$$\underset{{\lambda }_k\nearrow {\lambda }^{\ast }}{lim\; inf}\frac{I(s,p,{\lambda }_k)}{{\left({\lambda }^{\ast }-{\lambda }_k\right)}^{\frac{q}{2(s+1)+q}}}\ge \frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}}.$$

(61)

So far, we have completed the proof of Lemma 2. □

4. Proof of Theorems 2 and 3

In light of Propositions 1–4, Lemmas 1 and 2 established in Section 3, in this section, we shall provide the proof of Theorems 2 and 3. In order to do so, some descriptions are necessary. Suppose that $u_{\lambda }$ is a nonnegative minimizer of $I(s,p,\lambda )$, we define a normalized function

$$v_{\lambda }\left(x\right):={ϵ}_{\lambda }^{\frac{n}{2}}{u}_{\lambda }({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda }),x\in {\mathbb{R}}^n,$$

(62)

where ${ϵ}_{\lambda }$ and $x_{\lambda }$ are given by (34) and Proposition 2.

Proof of Theorem 2. 

Lemma 1 shows that for any $\lambda$ with $\lambda \nearrow {\lambda }^{\ast }$, the energy upper bound of $I(s,p,\lambda )$ in Theorem 2 holds. For the energy lower bound, we claim that for any $\lambda$ with $\lambda \nearrow {\lambda }^{\ast }$, $I(s,p,\lambda )$ satisfies the following:

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim\; inf}\frac{I(s,p,\lambda )}{{\left({\lambda }^{\ast }-\lambda \right)}^{\frac{q}{2(s+1)+q}}}\ge \frac{1}{{\lambda }^{\ast }(s+1)}{\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{\frac{2(s+1)}{2(s+1)+q}}+\alpha {\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{q}{2(s+1)+q}}.$$

(63)

Firstly, we claim that the energy lower bound in (63) holds for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$. If not, then there exists a sequence $\left\{{\lambda }_k^{\prime }\right\}$ with ${\lambda }_k^{\prime }\nearrow {\lambda }^{\ast }$, such that (63) is false. By repeating the proof of (61), one can prove that $\left\{{\lambda }_k^{\prime }\right\}$ has a subsequence (still denoted by $\left\{{\lambda }_k^{\prime }\right\}$), such that (61) holds, which then leads to a contradiction. Thus, (63) holds for any sequence $\left\{{\lambda }_k\right\}$ with ${\lambda }_k\nearrow {\lambda }^{\ast }$. Secondly, (63) is essentially the same for any $\lambda$ with $\lambda \nearrow {\lambda }^{\ast }$, which then gives the lower energy estimation of Theorem 2. □

Proof of Theorem 3. 

Repeating the proof procedures of Propositions 3 and 4 in Section 3, for any $\lambda$ with $\lambda \nearrow {\lambda }^{\ast }$, $v_{\lambda }$ defined by (62) satisfies the following for some $y_0\in {\mathbb{R}}^n$:

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}{v}_{\lambda }\left(x\right)=\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}{ϵ}_{\lambda }^{\frac{n}{2}}{u}_{\lambda }({ϵ}_{\lambda }x+{ϵ}_{\lambda }{x}_{\lambda })=\frac{w_p\left(\right|x-y_0\left|\right)}{\parallel w_p{\parallel }_{L^2}},$$

(64)

strongly in $H^1\left({\mathbb{R}}^n\right)$, and $w_p\left(x\right)$ is given by (5). Furthermore, similar to the estimation of (58), one can see that $y_0\equiv 0$. Repeating the proof of Proposition 4 and Lemma 2, one also deduces that the above ${ϵ}_{\lambda }$ and ${ϵ}_{\lambda }{x}_{\lambda }$ fulfill

$$\underset{\lambda \nearrow {\lambda }^{\ast }}{lim}\frac{{ϵ}_{\lambda }}{{\left({\lambda }^{\ast }-\lambda \right)}^{\frac{1}{2(s+1)+q}}}={\left(\frac{{\lambda }^{\ast }q\alpha }{2}\right)}^{-\frac{1}{2(s+1)+q}}$$

(65)

and

$${ϵ}_{\lambda }{x}_{\lambda }\to x_i\mathrm{as}\lambda \nearrow {\lambda }^{\ast },$$

(66)

where $x_i\in \mathcal{Z}$, and $x_i$ is one of the flattest global minima of potential $V\left(x\right)$. This completes the proof of Theorem 3. □