Existence and concentration of semiclassical states for nonlinear Schrodinger equations

In this article, we study the semilinear Schrodinger equation $$ -epsilon^2Delta u+ u+ V(x)u=f(u),quad uin H^1(mathbb{R}^N), $$ where $Ngeq 2$ and $epsilon>0$ is a small parameter. The function $V$ is bounded in $mathbb{R}^N$, $inf_{mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate isolated critical point. Under some conditions on f, we prove that as $epsilon o 0$, this equation has a solution which concentrates at the critical point of V.