On concentration of solution to a Schrödinger logarithmic equation with deepening potential well

In this work, we prove the existence of positive solution for the following class of problems− Δ u + λ V ( x ) u = u log u 2 , x ∈ R N ,u ∈ H 1 ( R N ) ,where λ >0 and V : R N → R is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C 1 functional with a convex lower semicontinuous functional, we prove that for each large enough λ >0, there exists a positive solution for the problem, and that, as λ →+ ∞ , such solutions converge to a positive solution of the limit problem defined on the domain Ω=int( V −1 ({0})).