2 01 0 N - Laplacian equations in R N with subcritical and critical growth without the Ambrosetti - Rabinowitz condition

Let Ω be a bounded domain in R . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type: (0.1) { −∆Nu = f (x, u) u ∈ W 1,2 0 (Ω) \ {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 (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in R 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 [11, 12]. 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.