Existence and concentration of semiclassical states for nonlinear Schrodinger equations

Details Existence and concentration of semiclassical states for nonlinear Schrodinger equations Existence and concentration of semiclassical states for nonlinear Schrodinger equations

2012-01-11

arxiv.org/abs/1201.2215v1

Mathematics

Analysis of PDEs

35 pages

Scientific paper

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

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