Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case

2023/05/24 by Cassani, Daniele, Liu, Zhisu, Romani, Giulio
#35A15 (Primary) #35B06 (Secondary) #35J62 #35Q55 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.15274

Abstract

We study the nonlinear Schrödinger equation for the s-fractional p-Laplacian strongly coupled with the Poisson equation in dimension two and with p=\frac2s, which is the limiting case for the embedding of the fractional Sobolev space W^s,p(ℝ²). We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.

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