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

In this paper, we study the following fractional Schrödinger equation (−Δ)su + λV(x)u = ulogu2, x ∈ ℝN, 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 → ∞.