Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth

TL;DR

This paper constructs multi-peak solutions for a singularly perturbed nonlinear elliptic problem with critical growth, demonstrating concentration around prescribed boundary-related points in possibly unbounded domains.

Contribution

It introduces a method to generate multi-peak solutions concentrating near any finite set of boundary maxima for critical growth problems.

Findings

Existence of multi-peak solutions concentrating near boundary maxima.

Solutions can be prescribed around any finite set of local maxima.

Applicable to unbounded domains with smooth boundaries.

Abstract

We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}\Delta u + u = f(u){\text{ in }}\Omega ,{\text{ }}u > 0{\text{ in }}\Omega ,{\text{ }}u = 0{\text{ on }}\partial \Omega , \] where $\Omega$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial \Omega \in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial \Omega$.