Multiplicity results for a class of quasilinear equations with exponential critical growth

TL;DR

This paper establishes the existence and multiplicity of positive solutions for a class of quasilinear elliptic equations with exponential critical growth, using variational methods and topological category theory.

Contribution

It introduces new multiplicity results for quasilinear equations with exponential critical growth, combining variational techniques with Lusternik-Schnirelman theory.

Findings

Proved existence of multiple positive solutions.

Applied variational methods to quasilinear equations.

Utilized Lusternik-Schnirelman category theory.

Abstract

In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations $$ \left \{ \begin{array}{lll} -\epsilon^{N}\Delta_{N} u + \left(1+\mu A(x) \right)\left| u\right|^{N-2}u= f(u)\,\,\,\, \mbox{ in } \, \RR^{N},\\ u > 0 \,\,\,\, \mbox{ in } \, \RR^{N}, \end{array} \right. $$ \\ where $\Delta_{N}$ is the N-Laplacian operator, $N \geq 2$, $f$ is a function with exponential critical growth, $\mu$ and $\epsilon$ are positive parameters and $A$ is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments and Lusternik-Schnirelman category theory .