Weighted Schrödinger-Kirchhoff type problem in dimension 2 with non-linear double exponential growth

Abstract

In this work, we study the weighted Kirchhoff problem {g(?B(?(x)|?u|2 + V(x)u2)dx) [?div(?(x)?u) + V(x)u] = f (x, u) in B u > 0 in B u = 0 on ?B, where B is the unit ball in R2, ?(x) = log e |x| , the singular logarithm weight in the Trudinger-Moser embedding, g is a continuous positive function on R+ and the potential V is a continuous positve function. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities. We prove the existence of non-trivial solutions via the critical point theory. In the critical case, the associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.