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

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.

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Liao, P. , Ping, R. and Chen, S. (2022) Erratum to “Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term” [Journal of Applied Mathematics and Physics (2022) 347-359]. Journal of Applied Mathematics and Physics, 10, 1297-1303. doi: 10.4236/jamp.2022.104091.

1. Introduction

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

2. Preliminary Results

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

${m}_{b}=\underset{u\in {M}_{b}}{\mathrm{inf}}E\left(u\right),$

where

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

and

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

We also define

${\omega }_{b}=\underset{v\in {W}_{b}}{\mathrm{inf}}F\left(v\right),$

where

${W}_{b}=\left\{v\in {H}_{V}^{1}\left({ℝ}^{N}\right):{‖{G}^{-1}\left(v\right)‖}_{{L}^{q+1}}=b\right\},\text{ }b>0,$

and

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

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

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

and

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

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

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

The proof of Lemma 2.4

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

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

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

$\begin{array}{l}\frac{\lambda }{2p}|{\int }_{{ℝ}^{N}}\left(|x|\ast {|{G}^{-1}\left({v}_{n}\right)|}^{p}\right){|{G}^{-1}\left({v}_{n}\right)|}^{p}-{\int }_{{ℝ}^{N}}\left(|x|\ast {|{G}^{-1}\left(v\right)|}^{p}\right){|{G}^{-1}\left(v\right)|}^{p}|\\ \le \frac{\lambda }{2p}\left(|{\int }_{{ℝ}^{N}}|x|\ast \left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right){|{G}^{-1}\left({v}_{n}\right)|}^{p}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+|{\int }_{{ℝ}^{N}}\left(|x|\ast {|{G}^{-1}\left(v\right)|}^{p}\right)\left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right)|\right)\\ \le C|{\left({\int }_{{ℝ}^{N}}{|{G}^{-1}\left({v}_{n}\right)|}^{pr}\right)}^{\frac{1}{r}}{\left({\int }_{{ℝ}^{N}}{\left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right)}^{r}\right)}^{\frac{1}{r}}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+C|{\left({\int }_{{ℝ}^{N}}{|{G}^{-1}\left(v\right)|}^{pr}\right)}^{\frac{1}{r}}{\left({\int }_{{ℝ}^{N}}{\left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right)}^{r}\right)}^{\frac{1}{r}}|,\end{array}$

and

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

where $r=\frac{2N}{2N-\mu }$. We know ${‖{v}_{n}-v‖}_{{H}_{V}^{1}\left({ℝ}^{N}\right)}\to 0$ 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]

$|{\int }_{{ℝ}^{N}}\left(|x|\ast {|{G}^{-1}\left({v}_{n}\right)|}^{p}\right){|{G}^{-1}\left({v}_{n}\right)|}^{p}-{\int }_{{ℝ}^{N}}\left(|x|\ast {|{G}^{-1}\left(v\right)|}^{p}\right){|{G}^{-1}\left(v\right)|}^{p}|\to 0,\text{\hspace{0.17em}}\text{ }\text{ }n\to +\infty .$

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

$\begin{array}{l}|\frac{1}{2t}{\int }_{{ℝ}^{N}}\text{ }\text{ }V\left(x\right)\left({G}^{-1}{\left(v+t\varphi \right)}^{2}-{G}^{-1}{\left(v\right)}^{2}\right)-{\int }_{{ℝ}^{N}}\frac{V\left(x\right){G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\varphi |\\ =|{\int }_{0}^{1}\text{ }\text{ }\text{d}s{\int }_{{ℝ}^{N}}\text{ }\text{ }V\left(x\right)\left(\frac{{G}^{-1}\left(v+ts\varphi \right)}{g\left({G}^{-1}\left(v+ts\varphi \right)\right)}-\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\right)\varphi |\\ \le {\int }_{0}^{1}\text{ }\text{ }\text{d}s{\left({\int }_{{ℝ}^{N}}\text{ }\text{ }V\left(x\right){|\frac{{G}^{-1}\left(v+ts\varphi \right)}{g\left({G}^{-1}\left(v+ts\varphi \right)\right)}-\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}|}^{2}\right)}^{\frac{1}{2}}{\int }_{0}^{1}\text{ }\text{ }\text{d}s{\left({\int }_{{ℝ}^{N}}\text{ }\text{ }V\left(x\right){\varphi }^{2}\right)}^{\frac{1}{2}}.\end{array}$

Using the definition of g and Lemma 1.1, we know

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

By the dominated convergence theorem

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

For the third term, we have

$\begin{array}{l}\lambda |{\int }_{{ℝ}^{N}}\frac{\left[{|x|}^{-\mu }\ast {|{G}^{-1}\left({v}_{n}\right)|}^{p}\right]{|{G}^{-1}\left({v}_{n}\right)|}^{p-2}{G}^{-1}\left({v}_{n}\right)}{g\left({G}^{-1}\left({v}_{n}\right)\right)}\varphi -{\int }_{{ℝ}^{N}}\frac{\left[{|x|}^{-\mu }\ast {|{G}^{-1}\left(v\right)|}^{p}\right]{|{G}^{-1}\left(v\right)|}^{p-2}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\varphi |\\ \le \lambda |{\int }_{{ℝ}^{N}}\frac{\left[{|x|}^{-\mu }\ast \left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right)\right]{|{G}^{-1}\left({v}_{n}\right)|}^{p-2}{G}^{-1}\left({v}_{n}\right)}{g\left({G}^{-1}\left({v}_{n}\right)\right)}\varphi |\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda |{\int }_{{ℝ}^{N}}\left({|x|}^{-\mu }\ast {|{G}^{-1}\left(v\right)|}^{p}\right)\left(\frac{{|{G}^{-1}\left({v}_{n}\right)|}^{p-2}{G}^{-1}\left({v}_{n}\right)}{g\left({G}^{-1}\left({v}_{n}\right)\right)}-\frac{{|{G}^{-1}\left(v\right)|}^{p-2}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\right)\varphi |\\ \le C{\int }_{{ℝ}^{N}}{\left({\left({|{G}^{-1}\left({v}_{n}\right)|}^{p}-{|{G}^{-1}\left(v\right)|}^{p}\right)}^{r}\right)}^{\frac{1}{r}}{\int }_{{ℝ}^{N}}{\left({|\frac{{|{G}^{-1}\left({v}_{n}\right)|}^{p-2}{G}^{-1}\left({v}_{n}\right)}{g\left({G}^{-1}\left({v}_{n}\right)\right)}\varphi |}^{r}\right)}^{\frac{1}{r}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+C{\left({\int }_{{ℝ}^{N}}{|{G}^{-1}\left(v\right)|}^{pr}\right)}^{\frac{1}{r}}{\left({\int }_{{ℝ}^{N}}{\left(\frac{{|{G}^{-1}\left({v}_{n}\right)|}^{p-2}{G}^{-1}\left({v}_{n}\right)}{g\left({G}^{-1}\left({v}_{n}\right)\right)}-\frac{{|{G}^{-1}\left(v\right)|}^{p-2}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\right)}^{\frac{p}{p-1}r}\right)}^{\frac{p-1}{pr}}{\left({\int }_{{ℝ}^{N}}{|\varphi |}^{pr}\right)}^{\frac{1}{pr}}.\end{array}$

and

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

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

where $r=\frac{2N}{2N-\mu }$, $2\le pr<{2}^{*}$. Since ${‖{v}_{n}-v‖}_{{H}_{V}^{1}\left({ℝ}^{N}\right)}\to 0$ 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

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

By Lemma 1.1

$\begin{array}{c}|〈{F}^{\prime }\left(v\right),\text{\hspace{0.17em}}\varphi 〉|=|{\int }_{{ℝ}^{N}}\text{ }\text{ }\nabla v\nabla \varphi +{\int }_{{ℝ}^{N}}\frac{V\left(x\right){G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\varphi \begin{array}{c}\text{ }\\ \text{ }\\ \text{ }\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{{ℝ}^{N}}\frac{\left[{|x|}^{-\mu }\ast {|{G}^{-1}\left(v\right)|}^{p}\right]{|{G}^{-1}\left(v\right)|}^{p-2}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\varphi |\\ \le C{‖v‖}_{{H}_{V}^{1}\left({ℝ}^{N}\right)}{‖\varphi ‖}_{{H}_{V}^{1}\left({ℝ}^{N}\right)}+C{‖v‖}_{{L}^{pr}\left({ℝ}^{N}\right)}^{p-1}{‖\varphi ‖}_{{L}^{pr}\left({ℝ}^{N}\right)}.\end{array}$

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 $q\ge {2}^{*}$ 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

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

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

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

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

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

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 ${u}_{n}⇀{u}_{b}$ 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

${\int }_{{ℝ}^{N}}\left(|x|\ast {|{u}_{n}|}^{p}\right){|{u}_{n}|}^{p}\to {\int }_{{ℝ}^{N}}\left(|x|\ast {|{u}_{b}|}^{p}\right){|{u}_{b}|}^{p},\text{ }n\to +\infty .$

Hence

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

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

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

$\begin{array}{c}|〈{{h}^{\prime }}_{q+1}\left(v\right),\phi 〉|=|{\int }_{{ℝ}^{N}}\frac{{|{G}^{-1}\left(v\right)|}^{q-1}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\phi |\\ \le C{\left({\int }_{{ℝ}^{N}}{|v|}^{q+1}\right)}^{\frac{q}{q+1}}{\left({\int }_{{ℝ}^{N}}{|\phi |}^{q+1}\right)}^{\frac{1}{q+1}}\\ \le C{‖\phi ‖}_{{H}_{V}^{1}\left({ℝ}^{N}\right)}.\end{array}$

then ${{h}^{\prime }}_{q+1}\left(v\right)\in {\left({H}_{V}^{1}\left({ℝ}^{N}\right)\right)}^{*}$.

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

and

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

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

$|〈{{h}^{\prime }}_{q+1}\left({v}_{n}\right)-{{h}^{\prime }}_{q+1}\left(v\right),\phi 〉|\to 0,\text{ }n\to +\infty .$

then ${h}_{q+1}\left(v\right)\in {C}^{1}\left({H}_{V}^{1}\left({ℝ}^{N}\right),\text{\hspace{0.17em}}ℝ\right)$ 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,

$\begin{array}{c}〈{F}^{\prime }\left(v\right),\phi 〉={\int }_{{ℝ}^{N}}\text{ }\text{ }\nabla v\nabla \phi +{\int }_{{ℝ}^{N}}\frac{V\left(x\right){G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\phi \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\lambda {\int }_{{ℝ}^{N}}\frac{\left[{|x|}^{-\mu }\ast {|{G}^{-1}\left(v\right)|}^{p}\right]{|{G}^{-1}\left(v\right)|}^{p-2}{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\phi .\end{array}$

Take limit $t\to 0$, we get $〈{F}^{\prime }\left({v}_{b}\right),v〉\ge 0$, by arbitrariness of v, one has $〈{F}^{\prime }\left({v}_{b}\right),-v〉\ge 0$. It follows that $〈{F}^{\prime }\left({v}_{b}\right),v〉=0$, 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 $〈{{h}^{\prime }}_{q+1}\left({v}_{b}\right),{v}^{\prime }〉=1$, for every $\phi \in {H}_{V}^{1}\left({ℝ}^{N}\right)$, let

$\psi =\phi -〈{{h}^{\prime }}_{q+1}\left({v}_{b}\right),\phi 〉{v}^{\prime }.$

Then $\psi \in \mathcal{N}\left({{h}^{\prime }}_{q+1}\left({v}_{b}\right)\right)$, it means $〈{F}^{\prime }\left({v}_{b}\right),\psi 〉=0$, i.e.

$〈{F}^{\prime }\left({v}_{b}\right),\phi 〉=〈{F}^{\prime }\left({v}_{b}\right),{v}^{\prime }〉〈{{h}^{\prime }}_{q+1}\left({v}_{b}\right),\phi 〉.$

Put $\beta =\beta \left(b\right)=〈{F}^{\prime }\left({v}_{b}\right),{v}^{\prime }〉$, we have

$〈{F}^{\prime }\left({v}_{b}\right),\phi 〉=\beta 〈{{h}^{\prime }}_{q+1}\left({v}_{b}\right),\phi 〉,$

n

Conflicts of Interest

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