Semi-classical states for the Choquard equation

Abstract

We study the nonlocal equation $$-{\epsilon }^2\Delta u_{\epsilon }+V{u}_{\epsilon }={\epsilon }^{-\alpha }(I_{\alpha }\ast |u_{\epsilon }{|}^p)|u_{\epsilon }{|}^{p-2}{u}_{\epsilon }\text{in RN},$$ where $N\ge 1$, $\alpha \in (0,N)$, $I_{\alpha }(x)=A_{\alpha }/|x{|}^{N-\alpha }$ is the Riesz potential and $\epsilon >0$ is a small parameter. We show that if the external potential $V\in C({\mathbb{R}}^N;[0,\infty ))$ has a local minimum and $p\in [2,(N+\alpha )/(N-2{)}_{+})$ then for all small $\epsilon >0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p>1+\max (\alpha ,\frac{\alpha +2}{2})/(N-2{)}_{+}$, or $p>2$ and ${\mathrm{liminf}}_{|x|\to \infty }V(x)|x{|}^2>0$, or $p=2$ and ${\inf }_{x\in {\mathbb{R}}^N}V(x)(1+|x{|}^{N-\alpha })>0$. Our assumptions on the decay of $V$ and admissible range of $p\ge 2$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.

28 pages, updated bibliography

References in corpus (2)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

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