On a class of nonhomogeneous elliptic problems involving exponential critical growth

Abstract

In this paper we establish the existence of solutions for elliptic equations of the form $-\text{div}(|\nabla u|^{n-2}\nabla u) + V(x)|u|^{n-2}u=g(x,u)+\lambda h$ in $\mathbb{R}^n$ with $n\geq2$. Here the potential $V(x)$ can change sign and the nonlinearity $g(x,u)$ is possibly discontinuous and may exhibit exponential growth. The proof relies on the application of a fixed point result and a version of the Trudinger-Moser inequality.