Existence and concentration of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via the penalization method

Abstract

We study positive solutions of the fractional logarithmic Schrödinger-Poisson system \[ \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^2+|u|^{q-2}u, & \text{in } \mathbb{R}^3, \varepsilon^{2α}(-Δ)^αϕ=u^2, & \text{in } \mathbb{R}^3, \end{cases} \] where \(\varepsilon>0\), \(3/4<α<1\), and \(4<q<6/(3-2α)\). We assume that the potential is continuous and nonnegative and attains its global minimum inside a bounded domain, with strictly larger values on the boundary. We work in the intersection of a weighted fractional Sobolev space and an Orlicz space, so that the logarithmic energy is finite and the associated functional is of class \(C^1\). For sufficiently small \(\varepsilon\), we obtain at least as many distinct positive solutions as the relative Lusternik-Schnirelmann category of the minimum set in a small neighborhood. The global maximum points of these solutions approach that minimum set as \(\varepsilon\to0\). The proof combines penalization, an autonomous Nehari-Pohozaev argument, a local natural constraint, and category theory.

arXiv commentWe have revised and rewritten the manuscript based on the reviewers' comments. 41 pages, comments are welcome