Ground state for Choquard equation with doubly critical growth nonlinearity

Abstract

In this paper we consider nonlinear Choquard equation \begin{equation*} -\Delta u+V(x)u=(I_\alpha*F(u))f(u)\quad {\rm in}\ \mathbb{R}^{N}, \end{equation*} where $V\in C(\mathbb{R}^N)$, $I_\alpha$ denotes the Riesz potential, $f(t)=|t|^{p-2}t+|t|^{q-2}t$ for all $t\in\mathbb{R}$, $N\geqslant5$ and $\alpha\in(0,N-4)$. Under suitable conditions on $V$, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., $p=(N+\alpha)/N,q=(N+\alpha)/(N-2)$, has a nonnegative ground state solution by variational methods.

Keywords

• choquard equation
• doubly critical nonlinearity
• ground state solution