Existence and concentration of positive solutions for a Schrodinger logarithmic equation

This article concerns with the existence and concentration of positive solutions for the following logarithmic elliptic equation - 2.u + V (x) u = u log u2, in RN, u. H1(RN), where > 0, N = 3 and V is a continuous function with a global minimum. Using variational method developed by Szulkin (Ann Inst H Poincar ' e Anal Non Lin ' eaire 3: 77- 109, 1986) for functionals which are sum of a C1 functional with a convex lower semicontinuous functional, we prove, for small enough > 0, the existence of positive solutions and concentration around of a minimum point of V, when approaches zero. We also study the cases when V is periodic or asymptotically periodic.