Ground state solutions for a nonlinear Choquard equation

Abstract

We discuss the existence of ground state solutions for the Choquard equation $$-\Delta u=(I_{\alpha }\ast F(u))F^{\prime }(u)\text{in }{\mathbb{R}}^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular the case of a homogeneous nonlinearity $F(u)=\frac{|u{|}^p}{p}$. The cases $N=2$ and $N\ge 3$ are treated differently in some steps. The solutions are found through a variational mountain pass strategy. The result presented are contained in the papers with arXiv ID 1212.2027 and 1604.03294

8 pages, survey of my talk at the Bru-To PDE's conference

References in corpus (1)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics