Ground State Solutions for Choquard Equation With Indefinite Potential and Critical Exponential Growth

Abstract

In this article, we study the following Choquard equation, $$\{\begin{array}{l}-\mathrm{\Delta }u+V(x)u=(I_{\alpha }*F(u))f(u),x\in {\mathbb{R}}^2,\\ u\in H^1({\mathbb{R}}^2),\end{array}$$ where $I_{\alpha }$ is the Riesz potential, $f:{\mathbb{R}}^2\to \mathbb{R}$ possessing critical exponential growth at infinity, $V$ is 1-periodic in $x_1$ , $x_2$ and zero lies in the gap of spectrum of $-\mathrm{\Delta }+V$ . We derive some detailed estimates to deal with difficulties arising from the strongly indefinite feature and the appearance of convolution term. Using a suitable variational framework based on the generalized Nehari manifold method, we reduce the indefinite problem to a definite case and succeed in finding a bounded Palais–Smale sequence. With the help of a proper auxiliary equation, we obtain a fine threshold to control the minimax level for above critical problem which allows us to restore the compactness. Existence of the ground state solutions is obtained via the concentration compactness argument and some delicate analyses.