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.

19 pages

References in corpus (1)

• Existence of Nontrivial Solutions for p-Laplacian Equations in {R}^{N}

Cited by in corpus (1)

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