Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

Abstract In this paper, we focus on the existence of solutions for the Choquard equation { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\al...

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Autores principales:	Jing Zhang, Qiongfen Zhang
Formato:	article
Lenguaje:	EN
Publicado:	SpringerOpen 2021
Materias:	Ground state solution Variable potential Choquard equation Critical point Analysis QA299.6-433
Acceso en línea:	https://doaj.org/article/a7ba79cadda24e71bc2a6a71c3ec82af
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spelling	oai:doaj.org-article:a7ba79cadda24e71bc2a6a71c3ec82af2021-12-05T12:07:31ZExistence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential10.1186/s13661-021-01576-91687-2770https://doaj.org/article/a7ba79cadda24e71bc2a6a71c3ec82af2021-12-01T00:00:00Zhttps://doi.org/10.1186/s13661-021-01576-9https://doaj.org/toc/1687-2770Abstract In this paper, we focus on the existence of solutions for the Choquard equation { − Δ u + V ( x ) u = ( I α ∗ \| u \| α N + 1 ) \| u \| α N − 1 u + λ \| u \| p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ where λ > 0 $\lambda >0$ is a parameter, α ∈ ( 0 , N ) $\alpha \in (0,N)$ , N ≥ 3 $N\ge 3$ , I α : R N → R $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ is the Riesz potential. As usual, α / N + 1 $\alpha /N+1$ is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if λ > λ ∗ $\lambda >\lambda _{}$ for some given number λ ∗ $\lambda _{}$ in three cases: (i) 2 < p < 4 N + 2 $2< p<\frac{4}{N}+2$ , (ii) p = 4 N + 2 $p=\frac{4}{N}+2$ , and (iii) 4 N + 2 < p < 2 ∗ $\frac{4}{N}+2< p<2^{*}$ . Our result improves the previous related ones in the literature.Jing ZhangQiongfen ZhangSpringerOpenarticleGround state solutionVariable potentialChoquard equationCritical pointAnalysisQA299.6-433ENBoundary Value Problems, Vol 2021, Iss 1, Pp 1-20 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Ground state solution Variable potential Choquard equation Critical point Analysis QA299.6-433
spellingShingle	Ground state solution Variable potential Choquard equation Critical point Analysis QA299.6-433 Jing Zhang Qiongfen Zhang Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
description	Abstract In this paper, we focus on the existence of solutions for the Choquard equation { − Δ u + V ( x ) u = ( I α ∗ \| u \| α N + 1 ) \| u \| α N − 1 u + λ \| u \| p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ where λ > 0 $\lambda >0$ is a parameter, α ∈ ( 0 , N ) $\alpha \in (0,N)$ , N ≥ 3 $N\ge 3$ , I α : R N → R $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ is the Riesz potential. As usual, α / N + 1 $\alpha /N+1$ is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if λ > λ ∗ $\lambda >\lambda _{}$ for some given number λ ∗ $\lambda _{}$ in three cases: (i) 2 < p < 4 N + 2 $2< p<\frac{4}{N}+2$ , (ii) p = 4 N + 2 $p=\frac{4}{N}+2$ , and (iii) 4 N + 2 < p < 2 ∗ $\frac{4}{N}+2< p<2^{*}$ . Our result improves the previous related ones in the literature.
format	article
author	Jing Zhang Qiongfen Zhang
author_facet	Jing Zhang Qiongfen Zhang
author_sort	Jing Zhang
title	Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
title_short	Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
title_full	Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
title_fullStr	Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
title_full_unstemmed	Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential
title_sort	existence of ground state solutions for a class of choquard equations with local nonlinear perturbation and variable potential
publisher	SpringerOpen
publishDate	2021
url	https://doaj.org/article/a7ba79cadda24e71bc2a6a71c3ec82af
work_keys_str_mv	AT jingzhang existenceofgroundstatesolutionsforaclassofchoquardequationswithlocalnonlinearperturbationandvariablepotential AT qiongfenzhang existenceofgroundstatesolutionsforaclassofchoquardequationswithlocalnonlinearperturbationandvariablepotential
_version_	1718372211555827712

Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

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