Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term

TL;DR

This paper proves the existence of positive solutions for a class of nonlinear elliptic equations involving the N-Laplacian, exponential nonlinearity, and gradient dependence, using an approximation scheme in finite-dimensional spaces.

Contribution

It introduces a novel approximation method in finite-dimensional normed spaces to handle gradient-dependent exponential nonlinearities, extending previous results to all dimensions greater than two.

Findings

Established positive solutions for N-Laplacian problems with exponential nonlinearity.

Extended prior results to all dimensions N>2.

Developed a new approximation scheme in finite-dimensional spaces.

Abstract

We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any $N>2$.