N-laplacian equations in RN with subcritical and critical growthwithout the ambrosetti-rabinowitz condition

Abstract

Let Ω be a bounded domain in RN. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: {equation presented} 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 (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N-Laplacian in RN 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 [9, 10, 21]. Examples of f and comparison with earlier assumptions in the literature are given.