Infinitely many solutions for nonhomogeneous Choquard equations

Abstract

In this paper, we study the following nonhomogeneous Choquard equation \begin{equation*} \begin{split} -\Delta u+V(x)u=(I_\alpha*|u|^p)|u|^{p-2}u+f(x),\qquad x\in \mathbb{R}^N, \end{split} \end{equation*} where $N\geq3,\alpha\in(0,N),p\in \big[\frac{N+\alpha}{N},\frac{N+\alpha}{N-2}\big)$, $I_\alpha$ denotes the Riesz potential and $f\neq 0$. By using a critical point theorem for non-even functionals, we prove the existence of infinitely many virtual critical points for two classes of potential $V$. To the best of our knowledge, this result seems to be the first one for nonhomogeneous Choquard equation on the existence of infinity many solutions.

Keywords

• choquard equation
• infinitely many solutions
• non-even functional
• variational methods