Standing Waves for nonlinear Schrodinger Equations involving critical growth

TL;DR

This paper constructs solutions to a nonlinear Schrödinger equation with critical growth that concentrate at specific points as a small parameter tends to zero, extending previous subcritical results.

Contribution

It introduces a method to handle critical growth nonlinearities in Schrödinger equations, completing the analysis beyond subcritical cases.

Findings

Solutions concentrate at local minima of the potential as epsilon approaches zero.

The method extends existing results to critical growth nonlinearities.

The paper provides conditions under which solutions exist and concentrate.

Abstract

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$ of the above problem which concentrates at an isolated component of positive local minimum points of $V$ as $\e\to 0$ under certain conditions on $f$. Our result completes the study made in some very recent works in the sense that, in those papers only the subcritical growth was considered