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arXiv · 2305.15274

Nonlocal planar Schr\"odinger-Poisson systems in the fractional Sobolev limiting case

Daniele Cassani·Zhisu Liu·Giulio Romani
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Abstract

We study the nonlinear Schr\"odinger 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 $W^{s,p}(\mathbb{R}^2)$. 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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Daniele Cassani, Zhisu Liu, Giulio Romani. 2023-05-24. Nonlocal planar Schr\"odinger-Poisson systems in the fractional Sobolev limiting case. https://doi.org/10.1016/j.jde.2023.11.018

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