Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

Abstract

Abstract In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ where N ≥ 3 $N\geq 3$ , 0 < μ < N $0<\mu <N$ , 2 N − μ N ≤ p < 2 N − μ N − 2 $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ , ∗ represents the convolution between two functions. We assume that the potential function V ( x ) $V(x)$ satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Keywords

• Choquard equation
• Ground state solutions
• Geometrically distinct solutions