Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method

TL;DR

This paper studies the existence and concentration of solutions to a $(p,N)$-Laplace equation with discontinuous nonlinearities and critical exponential growth, using penalization and variational methods.

Contribution

It introduces a penalization approach for locally Lipschitz functionals to handle irregular nonlinearities in the $(p,N)$-Laplace equation.

Findings

Established existence of solutions with discontinuous nonlinearities.

Demonstrated concentration behavior of solutions using variational and Moser iteration techniques.

Extended analysis to equations with critical exponential growth.

Abstract

In this paper, we investigate the existence and concentration of solutions to a $(p,N)$-Laplace equation in $\mathbb{R}^N$ involving a discontinuous nonlinearity and critical exponential growth. To establish the existence of solutions, we employ a penalization technique in the sense of Del Pino and Felmer adapted to a locally Lipschitz functional. Furthermore, by combining variational methods with Moser-type iteration techniques, we obtain the concentration behavior of the solutions. Our results contribute to the study of nonlinear elliptic problems with irregular nonlinearities and critical growth phenomena.