On a weighted N-Laplacian problem with double exponential nonlinearities

In this paper, we study the nonlinear elliptic problem -div(w(x)|∇ u|N-2∇ u)=f(x,u), u∈ W_01,N(B,w)(P) where B is the unit ball of R^N , N≥ 2 and w(x)=(/|x|)N-1 is the singular logarithm weight with the limiting exponent N-1 in the Trudinger-Moser embedding. We consider the problem (P) when the nonlinearity is sub-critical and critical with respect to a maximal growth of double exponent type and we prove the existence of positive solution by using Mountain Pass theorem without the Ambrosetti-Rabinowitz condition. When the nonlinearity is critical, we prove that the associated energy satisfies the Palais-Smale condition only to a given limit level and we prove that the min-max level is less than this limit.