Erratum to “Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term” [Journal of Applied Mathematics and Physics (2022) 347-359]

Peng Liao, Rui Ping, Shaoxiong Chen^*

School of Mathematics, Yunnan Normal University, Kunming, China.

DOI: 10.4236/jamp.2022.104091                

Abstract

The original online version of this article (Liao, P., Ping, R. and Chen, S. (2022) Positive Solutions for a Class of Quasilinear Schr?dinger Equations with Nonlocal Term. Journal of Applied Mathematics and Physics, 10, 347-359. https://doi.org/10.4236/jamp.2022.102027) needs some further amendments and clarification.

Keywords

Erratum

1. Introduction

where $N\ge 3$, $0<\mu <N$, $1\le q\le \frac{N}{N-2}$, $\frac{2N-\mu }{N}\le p<\frac{2N-\mu }{N-2}$, the function $V\in C\left({ℝ}^N\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>}{ℝ}^{+}\right)$, g is a ${\mathcal{C}}^1$ even function with $g^{\prime }\left(t\right)\le 0$ for all $t>0$, $g\left(0\right)=0$, ${\lim }_{t\to +\infty }g\left(t\right)=a$, $0<a<1$.

2. Preliminary Results

Next, we introduce some minimization with corresponding energy functional and define

$m_b=\underset{u\in M_b}{\inf }E\left(u\right)\mathrm{,}$

where

$M_b=\left\{u\in H^1\left({ℝ}^N\right):{\Vert u\Vert }_{L^{q+1}}=b\right\},b>0,$

and

$E\left(u\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[g^2\left(u\right){\left|\nabla u\right|}^2+V\left(x\right)u^2\right]-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[\left(\left|x\right|\ast {\left|u\right|}^p\right){\left|u\right|}^p\right].$

We also define

${\omega }_b=\underset{v\in W_b}{\inf }F\left(v\right)\mathrm{,}$

where

$W_b=\left\{v\in H_V^1\left({ℝ}^N\right):{\Vert G^{-1}\left(v\right)\Vert }_{L^{q+1}}=b\right\},b>0,$

and

$F\left(v\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla v\right|}^2+V\left(x\right)G^{-1}{\left(v\right)}^2\right)-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[\left(\left|x\right|\ast {\left|G^{-1}\left(v\right)\right|}^p\right){\left|G^{-1}\left(v\right)\right|}^p\right].$

Proof: For any $v\in W_b$, let $u=G^{-1}\left(v\right)$, from the definition of g, we get

${\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2={\displaystyle {\int }_{{ℝ}^N}}\frac{{\left|\nabla v\right|}^2}{g^2\left(G^{-1}\left(v\right)\right)}\le \frac{1}{a^2}{\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla v\right|}^2<+\infty ,$

and

${\displaystyle {\int }_{{ℝ}^N}}{u}^2\le {\displaystyle {\int }_{{ℝ}^N}}V\left(x\right)G^{-1}{\left(v\right)}^2<+\infty \mathrm{,}$

so $u\in M_b$. It follow that $F\left(v\right)=E\left(G^{-1}\left(v\right)\right)=E\left(u\right)\ge m_b$, hence ${\omega }_b\ge m_b$, moreover, for any $u\in M_b$, let $v=G\left(u\right)$, then $u=G^{-1}\left(v\right)$. We assume $E\left(u\right)<+\infty$, since $u\in H^1\left({ℝ}^N\right)$, $2<\frac{2Np}{2N-\mu }<2^{\ast }$, then $u\in L^{\frac{2Np}{2N-\mu }}\left({ℝ}^N\right)$. By Hardy-Little-Sobolev-inequality, we have

$\begin{array}{l}\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[g^2\left(u\right){\left|\nabla u\right|}^2+V\left(x\right)u^2\right]\\ =E\left(u\right)+\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|u\right|}^p\right){\left|u\right|}^p\\ \le E\left(u\right)+\frac{\lambda C}{2p}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^{\frac{2Np}{2N-\mu }}\right)}^{\frac{2N-\mu }{N}}<+\infty .\end{array}$ n

The proof of Lemma 2.4

Proof: (1) For any $v\in H_V^1\left({ℝ}^N\right)$, we have ${\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v\right)\right|}^{\frac{2Np}{2N-\mu }}\le C{\displaystyle {\int }_{{ℝ}^N}}{\left|v\right|}^{\frac{2Np}{2N-\mu }}<+\infty$, where $2\le \frac{2Np}{2N-\mu }<2^{*}$, similarly as the proof of Lemma2.3, by Hardy-Little-Sobolev-inequality, we have

$\begin{array}{c}F\left(v\right)=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla v\right|}^2+V\left(x\right)G^{-1}{\left(v\right)}^2\right)-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v\right)\right|}^p\right){\left|G^{-1}\left(v\right)\right|}^p\\ \le \frac{1}{2a^2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla v\right|}^2+V\left(x\right)v^2\right)+\frac{\lambda C}{2p}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v\right)\right|}^{\frac{2Np}{2N-\mu }}\right)}^{\frac{2N-\mu }{N}}\\ \le \frac{1}{2a^2}{\displaystyle {\int }_{{ℝ}^N}}\left({\left|\nabla v\right|}^2+V\left(x\right)v^2\right)+\frac{\lambda C}{2p}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|v\right|}^{\frac{2Np}{2N-\mu }}\right)}^{\frac{2N-\mu }{N}}<+\infty .\end{array}$

With the proof of continuity, note that F consist of three terms. By Lemma 1.1, we need to check the convolution term only. Using Hardy-Little-Sobolev-inequality

$\begin{array}{l}\frac{\lambda }{2p}\left|{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v_n\right)\right|}^p\right){\left|G^{-1}\left(v_n\right)\right|}^p-{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v\right)\right|}^p\right){\left|G^{-1}\left(v\right)\right|}^p\right|\\ \le \frac{\lambda }{2p}(\left|{\displaystyle {\int }_{{ℝ}^N}}\left|x\right|\ast \left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right){\left|G^{-1}\left(v_n\right)\right|}^p\right|\\ +\left|{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v\right)\right|}^p\right)\left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right)\right|)\\ \le C\left|{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v_n\right)\right|}^{pr}\right)}^{\frac{1}{r}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right)}^r\right)}^{\frac{1}{r}}\right|\\ +C\left|{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v\right)\right|}^{pr}\right)}^{\frac{1}{r}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right)}^r\right)}^{\frac{1}{r}}\right|,\end{array}$

and

${\left|{\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right|}^r\le C\left({\left|v_n\right|}^{pr}+{\left|v\right|}^{pr}\right)\mathrm{,}$

where $r=\frac{2N}{2N-\mu }$. We know ${\Vert v_n-v\Vert }_{H_V^1\left({ℝ}^N\right)}\to 0$ if $n\to +\infty$. So $\left\{v_n\right\}$ is bounded in $H_V^1\left({ℝ}^N\right)$. By Sobolev embedding theorem and Lemma 3.4 [22]

$\left|{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v_n\right)\right|}^p\right){\left|G^{-1}\left(v_n\right)\right|}^p-{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|G^{-1}\left(v\right)\right|}^p\right){\left|G^{-1}\left(v\right)\right|}^p\right|\to \mathrm{0,}n\to +\infty \mathrm{.}$

For (2) we consider the second and the third terms of the functional F, we see for $\varphi \in H_V^1\left({ℝ}^N\right)$, using Hölder inequality, we get

$\begin{array}{l}\left|\frac{1}{2t}{\displaystyle {\int }_{{ℝ}^N}}V\left(x\right)\left(G^{-1}{\left(v+t\varphi \right)}^2-G^{-1}{\left(v\right)}^2\right)-{\displaystyle {\int }_{{ℝ}^N}}\frac{V\left(x\right)G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\varphi \right|\\ =\left|{\displaystyle {\int }_0^1}\text{d}s{\displaystyle {\int }_{{ℝ}^N}}V\left(x\right)\left(\frac{G^{-1}\left(v+ts\varphi \right)}{g\left(G^{-1}\left(v+ts\varphi \right)\right)}-\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right)\varphi \right|\\ \le {\displaystyle {\int }_0^1}\text{d}s{\left({\displaystyle {\int }_{{ℝ}^N}}V\left(x\right){\left|\frac{G^{-1}\left(v+ts\varphi \right)}{g\left(G^{-1}\left(v+ts\varphi \right)\right)}-\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right|}^2\right)}^{\frac{1}{2}}{\displaystyle {\int }_0^1}\text{d}s{\left({\displaystyle {\int }_{{ℝ}^N}}V\left(x\right){\varphi }^2\right)}^{\frac{1}{2}}\mathrm{.}\end{array}$

Using the definition of g and Lemma 1.1, we know

$\begin{array}{c}{\left|\frac{G^{-1}\left(v+ts\varphi \right)}{g\left(G^{-1}\left(v+ts\varphi \right)\right)}-\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right|}^2\le {\left|G^{-1}\left(v+ts\varphi \right)+G^{-1}\left(v\right)\right|}^2\\ \le C\left({\left|G^{-1}\left(v+ts\varphi \right)\right|}^2+{\left|G^{-1}\left(v\right)\right|}^2\right)\\ \le C\left({\left|v+ts\varphi \right|}^2+{\left|v\right|}^2\right)\\ \le C\left({\left|v\right|}^2+{\left|\varphi \right|}^2\right)\mathrm{.}\end{array}$

By the dominated convergence theorem

$\left|{\displaystyle {\int }_0^1}\text{d}s{\displaystyle {\int }_{{ℝ}^N}}V\left(x\right)\left(\frac{G^{-1}\left(v+ts\varphi \right)}{g\left(G^{-1}\left(v+ts\varphi \right)\right)}-\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right)\varphi \right|\to \mathrm{0,}t\to 0.$

For the third term, we have

$\begin{array}{l}\lambda |{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v_n\right)\right|}^p\right]{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}\varphi -{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v\right)\right|}^p\right]{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\varphi |\\ \le \lambda \left|{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast \left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right)\right]{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}\varphi \right|\\ +\lambda \left|{\displaystyle {\int }_{{ℝ}^N}}\left({\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v\right)\right|}^p\right)\left(\frac{{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right)\varphi \right|\\ \le C{\displaystyle {\int }_{{ℝ}^N}}{\left({\left({\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right)}^r\right)}^{\frac{1}{r}}{\displaystyle {\int }_{{ℝ}^N}}{\left({\left|\frac{{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}\varphi \right|}^r\right)}^{\frac{1}{r}}\\ +C{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v\right)\right|}^{pr}\right)}^{\frac{1}{r}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left(\frac{{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right)}^{\frac{p}{p-1}r}\right)}^{\frac{p-1}{pr}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\varphi \right|}^{pr}\right)}^{\frac{1}{pr}}\mathrm{.}\end{array}$

and

${\left|{\left|G^{-1}\left(v_n\right)\right|}^p-{\left|G^{-1}\left(v\right)\right|}^p\right|}^r\le C\left({\left|v_n\right|}^{pr}+{\left|v\right|}^{pr}\right)\mathrm{,}$

${\left|\frac{{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right|}^{\frac{p}{p-1}r}\le C\left({\left|v_n\right|}^{pr}+{\left|v\right|}^{pr}\right)\mathrm{,}$

where $r=\frac{2N}{2N-\mu }$, $2\le pr<2^{*}$. Since ${\Vert v_n-v\Vert }_{H_V^1\left({ℝ}^N\right)}\to 0$ if $n\to +\infty$, $H_V^1\left({ℝ}^N\right)$ embedding into $L^r\left({ℝ}^N\right)$ is compact and $\left\{v_n\right\}$ is bounded in $H_V^1\left({ℝ}^N\right)$. Using Lemma 3.4 [22], we know

$\begin{array}{l}|{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v_n\right)\right|}^p\right]{\left|G^{-1}\left(v_n\right)\right|}^{p-2}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}\varphi \\ -{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v\right)\right|}^p\right]{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\varphi |\to \mathrm{0,}n\to +\infty \mathrm{.}\end{array}$

By Lemma 1.1

$\begin{array}{c}\left|\langle F^{\prime }\left(v\right),\mathrm{<mtext>\; \hspace{0.17em}\; </mtext>}\varphi \rangle \right|=|{\displaystyle {\int }_{{ℝ}^N}}\nabla v\nabla \varphi +{\displaystyle {\int }_{{ℝ}^N}}\frac{V\left(x\right)G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\varphi \begin{array}{c}\\ \\ \end{array}\\ -{\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v\right)\right|}^p\right]{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\varphi |\\ \le C{\Vert v\Vert }_{H_V^1\left({ℝ}^N\right)}{\Vert \varphi \Vert }_{H_V^1\left({ℝ}^N\right)}+C{\Vert v\Vert }_{L^{pr}\left({ℝ}^N\right)}^{p-1}{\Vert \varphi \Vert }_{L^{pr}\left({ℝ}^N\right)}\mathrm{.}\end{array}$

from Sobolev embedding theorem, we get $F^{\prime }\left(v\right)$ is a continuous linear functional on $H_V^1\left({ℝ}^N\right)$.n

3. Main Conclusion

Remark 3.1. From assumption of V, we know $H_V^1\left({ℝ}^N\right)$ embedding into $L^p\left({ℝ}^N\right)$ is compact. In the process of the proof of theorem 3.1, it is important for us to construct auxiliary function, then by implicit function theorem to prove it and lemma 3.4 [22] play a great role in this paper. Moreover, when $q\ge 2^{\mathrm{<mo>\; *\; </mo>}}$ is a open question for Equation (1.1), someone could do it if they are interested.

Proof of Theorem 3.1: Step 1: By the assumptions of (V₁) or (V₂), ${\omega }_b$ is achieved at some $0\le v_b\le W_b$ with $v_b\ne 0$.

Let $\left\{v_n\right\}\in W_b$ be a minimizing sequence for ${\omega }_b$. Set $u_n=G^{-1}\left(v_n\right)$, then $\left\{u_n\right\}\in M_b$ is a minimizing sequence for $m_b$. We can assume $u_n\ge 0$. It shows that $E\left(u_n\right)\to m_b$, so there exist $C>0$ such that

$\begin{array}{c}C\ge E\left(u_n\right)\\ =\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[g^2\left(u_n\right){\left|\nabla u_n\right|}^2+V\left(x\right)u_n^2\right]-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[\left(\left|x\right|\ast {\left|u_n\right|}^p\right){\left|u_n\right|}^p\right]\\ \ge \frac{a}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[{\left|\nabla u\right|}^2+V\left(x\right)u_n^2\right]-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|u_n\right|}^p\right){\left|u_n\right|}^p.\end{array}$

By Hölder inequality and Hardy-Little-Sobolev-inequality,

$\begin{array}{c}{\displaystyle {\int }_{{ℝ}^N}}\left[\left(\left|x\right|\ast {\left|u_n\right|}^p\right){\left|u_n\right|}^p\right]\le {\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u_n\right|}^{pr}\right)}^{\frac{2}{r}}\\ \le {\left({\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u_n\right|}^2\right)}^{\frac{\theta pr}{2}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u_n\right|}^{q+1}\right)}^{\frac{\left(1-\theta \right)pr}{q+1}}\right)}^{\frac{2}{r}}\\ ={\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u_n\right|}^2\right)}^{\theta p}{b}^{\frac{2\left(1-\theta \right)p}{q+1}}\\ \le b^{\frac{2p-\theta p+\theta pq-q-1}{q+1}}\left(\epsilon {\displaystyle {\int }_{{ℝ}^N}}{\left|u_n\right|}^2+C\left(\epsilon \right)b\right).\end{array}$

where $\theta =\frac{2\left(q+1\right)-2pr}{\left(q-1\right)pr}$, $0<\theta p<1$, $r=\frac{2N}{2N-\mu }$, $\epsilon >0$, $C\left(\epsilon \right)={\left(\frac{\epsilon }{\theta p}\right)}^{-\frac{\theta p}{1-\theta p}}\left(1-\theta p\right)$, then

$\begin{array}{c}C\ge E\left(u_n\right)\\ \ge \frac{a}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[{\left|\nabla u_n\right|}^2+V\left(x\right)u_n^2\right]-\frac{\lambda b^{\frac{2p-\theta p+\theta pq-q-1}{q+1}}}{2p}\left(\epsilon {\displaystyle {\int }_{{ℝ}^N}}V\left(x\right)u_n^2+C\left(\epsilon \right)b\right)\\ \ge \left(\frac{a}{2}-\frac{\lambda \epsilon b^{\frac{2p-\theta p+\theta pq-q-1}{q+1}}}{2p}\right)\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u_n\right|}^2+V\left(x\right){\left|u_n\right|}^2\right)-\frac{\lambda }{2p}C\left(\epsilon \right)b^{\frac{2p-\theta p+\theta pq}{q+1}}.\end{array}$

Taking $\epsilon >0$ small enough such that $\frac{a}{2}-\frac{\lambda \epsilon b^{\frac{2p-\theta p+\theta pq-q-1}{q+1}}}{2p}>0$. It implies that $u_n\left(x\right)$ is bounded in $H_V^1\left({ℝ}^N\right)$. By the compact embedding result from $H_V^1\left({ℝ}^N\right)$ into $L^r\left({ℝ}^N\right)$ for $2\le r<2^{\ast }$. We may assume that $u_n\rightharpoonup u_b$ in $H_V^1\left({ℝ}^N\right)$, $u_n\to u_b$ in $L^r\left({ℝ}^N\right)$ for $2\le r<2^{\ast }$ and $u_n\left(x\right)\to u_b\left(x\right)$ a.e $x\in {ℝ}^N$. Hence $u_b\in M_b$, since $u_n\ge 0$, $u_b\ge 0$ and $u_b\ne 0$. Similarly as the proof of Lemma 2.4 (1), we have

${\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|u_n\right|}^p\right){\left|u_n\right|}^p\to {\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|u_b\right|}^p\right){\left|u_b\right|}^p\mathrm{,}n\to +\infty \mathrm{.}$

Hence

$\begin{array}{c}{m}_b=\underset{n\to \infty }{\lim }E\left(u_n\right)\\ \ge \underset{n\to \infty }{\lim \inf }\left\{\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}\left[g^2\left(u_n\right){\left|\nabla u_n\right|}^2+V\left(x\right)u_n^2\right]-\frac{\lambda }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left(\left|x\right|\ast {\left|u_n\right|}^p\right){\left|u_n\right|}^p\right\}\\ \ge E\left(u_b\right).\end{array}$

Step 2: Set $h_{q+1}\left(v\right)=\frac{1}{q+1}{\displaystyle {\int }_{{ℝ}^N}}{\left|G^{-1}\left(v\left(x\right)\right)\right|}^{q+1}$ for $2\le q+1<2^{*}$, then $h_{q+1}\left(v\right)\in C^1\left(H_V^1\left({ℝ}^N\right)\mathrm{,}ℝ\right)$.

In fact, for any $\phi \in H_V^1\left({ℝ}^N\right)$, by Lemma 1.1 and Hölder’s inequality, we have

$\begin{array}{c}\left|\langle {h^{\prime }}_{q+1}\left(v\right),\phi \rangle \right|=\left|{\displaystyle {\int }_{{ℝ}^N}}\frac{{\left|G^{-1}\left(v\right)\right|}^{q-1}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\phi \right|\\ \le C{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|v\right|}^{q+1}\right)}^{\frac{q}{q+1}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\phi \right|}^{q+1}\right)}^{\frac{1}{q+1}}\\ \le C{\Vert \phi \Vert }_{H_V^1\left({ℝ}^N\right)}.\end{array}$

then ${h^{\prime }}_{q+1}\left(v\right)\in {\left(H_V^1\left({ℝ}^N\right)\right)}^{\mathrm{<mo>\; *\; </mo>}}$.

$\begin{array}{l}\left|\langle {h^{\prime }}_{q+1}\left(v_n\right)-{h^{\prime }}_{q+1}\left(v\right)\mathrm{,}\phi \rangle \right|\\ =\left|{\displaystyle {\int }_{{ℝ}^N}}\left(\frac{{\left|G^{-1}\left(v_n\right)\right|}^{q-1}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{q-1}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right)\phi \right|\\ \le {\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\frac{{\left|G^{-1}\left(v_n\right)\right|}^{q-1}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{q-1}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right|}^{\frac{q+1}{q}}\right)}^{\frac{q}{q+1}}{\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\phi \right|}^{q+1}\right)}^{\frac{1}{q+1}}\mathrm{.}\end{array}$ (1)

and

${\left|\frac{{\left|G^{-1}\left(v_n\right)\right|}^{q-1}{G}^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{{\left|G^{-1}\left(v\right)\right|}^{q-1}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\right|}^{\frac{q+1}{q}}\le C\left({\left|v_n\right|}^{q+1}+{\left|v\right|}^{q+1}\right)\mathrm{,}$

Since $v_n\to v$ in $H_V^1\left({ℝ}^N\right)$, $H_V^1\left({ℝ}^N\right)$ embedding into $L^r\left({ℝ}^N\right)$ is compact and $\left\{v_n\right\}$ is bounded in $H_V^1\left({ℝ}^N\right)$, where $2\le r<2^{*}$. By $2\le q+1<2^{*}$ and Lemma 3.4 [22], we have

$\left|\langle {h^{\prime }}_{q+1}\left(v_n\right)-{h^{\prime }}_{q+1}\left(v\right)\mathrm{,}\phi \rangle \right|\to \mathrm{0,}n\to +\infty \mathrm{.}$

then $h_{q+1}\left(v\right)\in C^1\left(H_V^1\left({ℝ}^N\right)\mathrm{,\; <mtext>\; \hspace{0.17em}\; </mtext>}ℝ\right)$ for $2\le q+1<2^{*}$.

Step 3: For any $b\ge 0$, there exist $\beta \left(b\right)\in ℝ$ such that $0<u_b=G^{-1}\left(v_b\right)\in M_b$ is a weak solution of Equation (1.1) with $\lambda =\lambda \left(b\right)$. In fact, by lemma2.4,

$\begin{array}{c}\langle F^{\prime }\left(v\right),\phi \rangle ={\displaystyle {\int }_{{ℝ}^N}}\nabla v\nabla \phi +{\displaystyle {\int }_{{ℝ}^N}}\frac{V\left(x\right)G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\phi \\ -\lambda {\displaystyle {\int }_{{ℝ}^N}}\frac{\left[{\left|x\right|}^{-\mu }\ast {\left|G^{-1}\left(v\right)\right|}^p\right]{\left|G^{-1}\left(v\right)\right|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}\phi .\end{array}$

Take limit $t\to 0$, we get $\langle F^{\prime }\left(v_b\right)\mathrm{,}v\rangle \ge 0$, by arbitrariness of v, one has $\langle F^{\prime }\left(v_b\right)\mathrm{,}-v\rangle \ge 0$. It follows that $\langle F^{\prime }\left(v_b\right)\mathrm{,}v\rangle =0$, for every $v\in \mathcal{N}\left({h^{\prime }}_{q+1}\left(v_b\right)\right)$. Set $v^{\prime }\in H_V^1\left({ℝ}^N\right)$ be such that $\langle {h^{\prime }}_{q+1}\left(v_b\right)\mathrm{,}{v}^{\prime }\rangle =1$, for every $\phi \in H_V^1\left({ℝ}^N\right)$, let

$\psi =\phi -\langle {h^{\prime }}_{q+1}\left(v_b\right)\mathrm{,}\phi \rangle v^{\prime }\mathrm{.}$

Then $\psi \in \mathcal{N}\left({h^{\prime }}_{q+1}\left(v_b\right)\right)$, it means $\langle F^{\prime }\left(v_b\right)\mathrm{,}\psi \rangle =0$, i.e.

$\langle F^{\prime }\left(v_b\right)\mathrm{,}\phi \rangle =\langle F^{\prime }\left(v_b\right)\mathrm{,}{v}^{\prime }\rangle \langle {h^{\prime }}_{q+1}\left(v_b\right)\mathrm{,}\phi \rangle \mathrm{.}$

Put $\beta =\beta \left(b\right)=\langle F^{\prime }\left(v_b\right)\mathrm{,}{v}^{\prime }\rangle$, we have

$\langle F^{\prime }\left(v_b\right)\mathrm{,}\phi \rangle =\beta \langle {h^{\prime }}_{q+1}\left(v_b\right)\mathrm{,}\phi \rangle \mathrm{,}$

n

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References