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

Abstract

We study the nonlinear Schrödinger equation for the $s-$fractional $p-$Laplacian strongly coupled with the Poisson equation in dimension two and with $p=\frac{2}{s}$, which is the limiting case for the embedding of the fractional Sobolev space $W^{s,p}({\mathbb{R}}^2)$. 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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Cited by in corpus (3)

• 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