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Research Article | Open Access

Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential

School of Mathematics and Physics, University of Science and Technology Beijing, Xueyuan Road 30, Haidian, Beijing 100083, China
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Abstract

In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential

−Δu+V(ϵx)u=λ(Iα∗|u|p)|u|p−1+ulog⁡u2inR3,

where u∈H1(R3), ϵ>0, V is a continuous function with a global minimum, and Coulomb type energies with 0<α<3 and p≥1. We explore the existence of local positive solutions without the functional having to be a combination of a C1 functional and a convex semicontinuous functional, as is required in the global case.

Keywords

variational methodlogarithmic Schrödinger equationmultiple solutionsCoulomb potential
CLC number: 35Q55, 35A15, 35J60, 35B09

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Communications in Analysis and Mechanics
Volume 16 Issue 3,
September 2024
Pages 487-508
DOI: 10.3934/cam.2024023

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Dong F. Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. https://doi.org/10.3934/cam.2024023

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Received: 29 November 2023
Revised: 19 January 2024
Accepted: 06 March 2024
Published: 15 September 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)