Schrödinger-Poisson systems with zero mass in the Sobolev limiting case

Abstract

We study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted $N$-Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is considered in ${\mathbb{R}}^N$, the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a sign-changing logarithm. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation where the logarithm is uniformly approximated by polynomial kernels.

References in corpus (7)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• A guide to the Choquard equation
• Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
• The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate
• Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
• Choquard equations with critical exponential nonlinearities in the zero mass case
• The Choquard logarithmic equation involving a nonlinearity with exponential growth

Cited by in corpus (2)

• Choquard equations with critical exponential nonlinearities in the zero mass case
• Nonlinear Schrödinger-Poisson systems in dimension two: the zero mass case