On existence and concentration of solutions for fractional logarithmic Schrödinger equation with steep potential well

¹ College of Mathematics, Jilin University, Changchun 130012, PR China
² Department of Ecological and Biological Sciences, University of Tuscia, Largo dell’Universitá 01100, Viterbo, Italy
³ Departament de Matemàtiques, Universitat Politècnica de Catalunya, Avinguda Diagonal 647, 08028 Barcelona, Catalunya, Spain & College of Mathematics, Jilin University, Changchun 130012, PR China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 3 July 2025
Accepted: 10 December 2025

Abstract

In this paper, we study the following fractional Schrödinger equation

(−Δ)^su + λV(x)u = ulogu², x ∈ ℝ^N,

where 0 < s < 1, λ > 0 and V : ℝⁿ → ℝ is a measurable potential satisfying some assumptions. Since the logarithmic term is singular at the origin, the energy functional is not of class 𝒞¹. To overcome this difficulty, the nonsmooth critical point theory developed by Szulkin is used. Employing variational methods, the existence of a nonnegative least energy solution for the problem is established for sufficient large λ's. In addition, we prove that such solutions converge to a least energy solution of the limit problem defined on a bounded domain as λ → ∞.