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

Abstract

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.

Keywords

• Ground state solution
• Variable potential
• Choquard equation
• Critical point