Elliptic equations in dimension 2 with double exponential nonlinearities

Abstract

A boundary value problem on the unit disk in \(\mathbb R^2\) is considered, involving an elliptic operator with a singular weight of logarithmic type and nonlinearities which are subcritical or critical with respect to the associated gradient norm. The existence of non-trivial solutions is proved, relying on variational methods. In the critical case, the associated energy functional is non-compact. A suitable asymptotic condition allows to avoid the non-compactness levels of the functional.