Non-autonomous weighted elliptic equations with double exponential growth

We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u > 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.