Existence and concentration of semiclassical states for nonlinear Schrodinger equations

In this paper, we study the following semilinear Schrödinger equation $-ε^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}),$ where $N\geq 2$ and $ε>0$ 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 $ε\rightarrow 0,$ this equation has a solution which concentrates at the critical point of $V$.}

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