### 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.

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.

#### Keywords

Critical growth; Trudinger-Moser inequality; fixed point result; discontinuous nonlinearity

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