Existence and concentration of semiclassical states for nonlinear Schrodinger equations

TL;DR

This paper investigates the existence and concentration behavior of semiclassical states for a nonlinear Schrödinger equation with a potential having possibly degenerate critical points, showing solutions concentrate at these points as the semiclassical parameter tends to zero.

Contribution

It establishes the existence and concentration of solutions for the nonlinear Schrödinger equation with degenerate critical points of the potential, extending previous results to more general potentials.

Findings

Solutions concentrate at critical points of V as epsilon approaches zero.

Existence of solutions under certain conditions on the nonlinearity f.

Concentration phenomena occur even with degenerate critical points.

Abstract

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 $V$.}