Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
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| Format: | Preprint |
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2023
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| _version_ | 1866918101952495616 |
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| author | Cassani, Daniele Liu, Zhisu Romani, Giulio |
| author_facet | Cassani, Daniele Liu, Zhisu Romani, Giulio |
| contents | 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 $W^{s,p}(\mathbb{R}^2)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_15274 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case Cassani, Daniele Liu, Zhisu Romani, Giulio Analysis of PDEs 35A15 (Primary), 35R11, 35J62, 35Q55, 35B06 (Secondary) 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 $W^{s,p}(\mathbb{R}^2)$. 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. |
| title | Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case |
| topic | Analysis of PDEs 35A15 (Primary), 35R11, 35J62, 35Q55, 35B06 (Secondary) |
| url | https://arxiv.org/abs/2305.15274 |