Multi-peak solutions for a class of degenerate elliptic equations

TL;DR

This paper proves the existence of multi-peak solutions for certain degenerate quasilinear elliptic equations using penalization methods, addressing challenges related to the limit equation and solution concentration.

Contribution

It extends the penalization approach to degenerate elliptic equations, providing new insights into multi-peak solutions under natural growth conditions.

Findings

Existence of multi-peak solutions established

Analysis of solution concentration using Pucci-Serrin identity

Addresses difficulties in the limit equation properties

Abstract

By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi-spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci-Serrin type identity plays an important role.