Existence and concentration of positive solutions for a Schrödinger logarithmic equation

Abstract

This article concerns with the existence and concentration of positive solutions for the following logarithmic elliptic equation $$\begin{aligned} \left\{ \begin{array}{lc} -{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{} \text{ in } \quad \mathbb {R}^{N}, \\ u \in H^1(\mathbb {R}^{N}), &{} \; \\ \end{array}\right. \end{aligned}$$ where $$\epsilon >0, N \ge 3$$ and V is a continuous function with a global minimum. Using variational method developed by Szulkin (Ann Inst H Poincare Anal Non Lineaire 3:77–109, 1986) for functionals which are sum of a $$C^1$$ functional with a convex lower semicontinuous functional, we prove, for small enough $$\epsilon >0$$ , the existence of positive solutions and concentration around of a minimum point of V, when $$\epsilon $$ approaches zero. We also study the cases when V is periodic or asymptotically periodic.