Concentration-Compactness principle for Trudinger-Moser inequalities with logarithmic weights and their applications

In this paper, we establish a sharp concentration-compactness principle associated with the Trudinger-Moser inequality on Sobolev spaces with logarithmic weights. As applications, we establish the existence of ground state solutions to the following equation with critical double exponential nonlinearity {-div(vertical bar Vu vertical bar(N-2) del uv(x)) = f (x, u), in B-1 (0), u > 0, in B-1 (0), u = 0, on partial derivative B-1 (0). where nu(x) = vertical bar log(e/vertical bar x vertical bar)vertical bar(N-1) is the logarithmic weight, the nonlinear term f (x, u) is continuous, radial in x is an element of B-1(0) and has critical double exponential growth which ivuN' behaves like exp(ee(NuN')). (C) 2020 Elsevier Ltd. All rights reserved.