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On a planar Schrödinger–Poisson system involving a non-symmetric potential

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  05 December 2022

Riccardo Molle Open the ORCID record for Riccardo Molle [Opens in a new window]  and
Andrea Sardilli
Riccardo Molle
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy (molle@mat.uniroma2.it; andreasardilli6@gmail.com)
Andrea Sardilli
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy (molle@mat.uniroma2.it; andreasardilli6@gmail.com)
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Abstract

We prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form

\[ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \]
where $p\ge 4$, $\gamma,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.

Keywords

nonlinear Schrödinger–Poisson systemsplanar casepositive solutions

MSC classification

Primary: 35J50: Variational methods for elliptic systems 35J47: Second-order elliptic systems 35J10: Schrödinger operator 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1133 - 1146
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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