Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities

In this paper, we prove the existence of nontrivial solutions forthe following weighted problem without the Ambrosetti-Rabinowitzcondition:-div(sigma(x)|del u|N-2 del u) =f(x,u) andu >0 inB,u= 0 on partial derivative B, whereBis the unit ball ofRN,sigma(x) =(log(e|x|))N-1is the singularlogarithmic weight in the Trudinger-Moser embedding. The nonlinearity isa critical or subcritical growth in view of Trudinger-Moser inequalities. Inorder to obtain the existence result, we used minimax techniques combinedwith the Trudinger-Moser inequality. In the critical case, the associatedenergy does not satisfy the condition of compactness. We provide a newcondition for growth and we stress its importance to avoid compactnesslevel