Choquard equations under confining external potentials

Abstract

We consider the nonlinear Choquard equation $$-\Delta u+Vu=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u\text{ in }{\mathbb{R}}^N$$ where $N\ge 1$, $I_{\alpha }$ is the Riesz potential integral operator of order $\alpha \in (0,N)$ and $p>1$. If the potential $V\in C({\mathbb{R}}^N;[0,+\infty ))$ satisfies the confining condition $$\underset{|x|\to +\infty }{\mathrm{liminf}}\frac{V(x)}{1+|x{|}^{\frac{N+\alpha }{p}-N}}=+\infty ,$$ and $\frac{1}{p}>\frac{N-2}{N+\alpha }$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p\ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Pohožaev identity that we establish.

21 pages

References in corpus (4)

• Gravitation and quantummechanical localization of macroobjects
• Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent
• Nodal solutions for the Choquard equation
• Least action nodal solutions for the quadratic Choquard equation

Cited by in corpus (5)

• Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent
• Standing waves with a critical frequency for nonlinear Choquard equations
• Schrödinger-Newton-Hooke system in higher dimensions. Part I: Stationary states
• Saddle solutions for the fractional Choquard equation
• Positive bound states to nonlinear Choquard equations in the presence of nonsymmetric potentials