Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent

Abstract

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $-\Delta u+V_{\mu ,\nu }u=(I_{\alpha }\ast |u{|}^{\frac{N+\alpha }{N}}){|u|}^{\frac{\alpha }{N}-1}u$, in ${\mathbb{R}}^N$ where $N\ge 3$, $V_{\mu ,\nu }:{\mathbb{R}}^N\to \mathbb{R}$ is an external potential defined for $\mu ,\nu \in \mathbb{R}$ and $x\in {\mathbb{R}}^N$ by $V_{\mu ,\nu }(x)=1-\mu /({\nu }^2+|x{|}^2)$ and $I_{\alpha }:{\mathbb{R}}^N\to 0$ is the Riesz potential for $\alpha \in (0,N)$, we exhibit two thresholds ${\mu }_{\nu },{\mu }^{\nu }>0$ such that the equation admits a positive ground state solution if and only if ${\mu }_{\nu }<\mu <{\mu }^{\nu }$ and no ground state solution exists for $\mu <{\mu }_{\nu }$. Moreover, if $\mu >\max \left\{{\mu }^{\nu },\frac{N^2(N-2)}{4(N+1)}\right\}$, then equation still admits a sign changing ground state solution provided $N\ge 4$ or in dimension $N=3$ if in addition $\frac{3}{2}<\alpha <3$ and ker$(-\Delta +V_{\mu ,\nu })=\{0\}$, namely in the non-resonant case.

References in corpus (3)

• A guide to the Choquard equation
• Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent
• Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency

Cited by in corpus (6)

• Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent
• Existence of solutions for critical Choquard equations via the concentration compactness method
• Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
• Choquard equations with critical exponential nonlinearities in the zero mass case
• Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
• Nonlinear Schrödinger-Poisson systems in dimension two: the zero mass case