Ground state for Choquard equation with doubly critical growth nonlinearity - Fuyi Li; Lei Long; Yongyan Huang; Zhanping Liang (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
In this paper we consider nonlinear Choquard equation \begin{equation*} -Δ u+V(x)u=(I_α*F(u))f(u)\quad {\rm in}\ \mathbb{R}^{N}, \end{equation*} where $V\in C(\mathbb{R}^N)$, $I_α$ denotes the Riesz potential, $f(t)=|t|^{p-2}t+|t|^{q-2}t$ for all $t\in\mathbb{R}$, $N\geqslant5$ and $α\in(0,N-4)$. Under suitable conditions on $V$, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., $p=(N+α)/N,q=(N+α)/(N-2)$, has a nonnegative ground state solution by variational methods.