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

Abstract

<p>In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\Delta u+V(\epsilon x) u = \lambda (I_\alpha * |u|^p)|u|^{p-1}+u \log u^2 \text { in } \mathbb{R}^3, \end{equation*} $\end{document} </tex-math></disp-formula></p><p>where $ u \in H^1(\mathbb{R}^3) $, $ \epsilon &gt; 0 $, $ V $ is a continuous function with a global minimum, and Coulomb type energies with $ 0 &lt; \alpha &lt; 3 $ and $ p \geq 1 $. We explore the existence of local positive solutions without the functional having to be a combination of a $ C^1 $ functional and a convex semicontinuous functional, as is required in the global case.</p>