Existence and multiplicity of solutions to equations of $N-$Laplacian type with critical exponential growth in $\mathbb{R}^{N}$

TL;DR

This paper establishes the existence and multiplicity of solutions for N-Laplacian equations with critical exponential growth in R^N, using variational methods despite potential compactness issues.

Contribution

It introduces new variational techniques to prove solutions without relying on the Ambrosetti-Rabinowitz condition for N-Laplacian equations.

Findings

Proved existence of weak solutions under critical exponential growth.

Established multiplicity of solutions using minimax and Ekeland variational principles.

Extended results to nonlinearities not satisfying the Ambrosetti-Rabinowitz condition.

Abstract

In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions. Finally, we will prove the above results when our nonlinearity doesn't satisfy the well-known Ambrosetti-Rabinowitz condition and thus derive the existence and multiplicity of solutions for a much wider class of nonlinear terms $f$.