On a weighted N- Laplacian problem with double exponential non linear growth

Abstract

\begin{abstract}
In this paper, we study the nonlinear elliptic problem
$$-\textmd{div} (w(x)|\nabla u|^{N-2} \nabla u) = f(x,u),~~u\in W_{0}^{1,N}(B_{1},w) \eqno(P)$$
where $B_{1}$ the unit ball of $\mathbb{R^{N}}$, $N\geq2$ and $ w(x)=\big(\log \frac{e}{|x|}\big)^{N-1}$
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-Rabionowitz 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.\end{abstract}