N-Laplacian equations in $\mathbb{R}^{N}$ with subcritical and critical growth without the Ambrosetti-Rabinowitz condition

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$. In this paper, we consider the following nonlinear elliptic equation of $N$-Laplacian type: $-\Delta_{N}u=f(x,u)$ where $u\in W_{0}^{1,2}\{0}$ when $f$ is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to the above equation without the Ambrosetti-Rabinowitz $(AR)$ condition. Earlier works in the literature on the existence of nontrivial solutions to $N-$Laplacian in $\mathbb{R}^{N}$ when the nonlinear term $f$ has the exponential growth only deal with the case when $f$ satisfies the $(AR)$ condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami \cite{Ce1, Ce2}. This approach can also be used to yield an existence result for the $p$-Laplacian equation ($1<p<N$) in the subcritical polynomial growth case.