Boundary value problems for Choquard equations

Abstract

We consider the following nonlinear Choquard equation $-\Delta u+Vu=(I_{\alpha }\ast {\left|u\right|}^p){\left|u\right|}^{p-2}u\text{in}\Omega \subset R^N,$ where $N\ge 2$, $p\in (1,+\infty )$, $V\left(x\right)$ is a continuous radial function such that ${inf}_{x\in \Omega }V>0$ and $I_{\alpha }\left(x\right)$ is the Riesz potential of order $\alpha \in (0,N)$. Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when $\Omega$ is an annulus, or an exterior domain of the form $R^N\setminus \overline{B_r\left(0\right)}$. We also provide a nonexistence result: if $p\ge \frac{N+\alpha }{N-2}$ the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting $\alpha \to 0^{+}$ we recover existence results for the corresponding local problem with power-type nonlinearity.