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

Abstract

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(|∇u|N−2∇uν(x))=f(x,u),inB1(0),u>0,inB1(0),u=0,on∂B1(0).where ν(x)=|log(e|x|)|N−1 is the logarithmic weight, the nonlinear term f(x,u) is continuous, radial in x∈B1(0) and has critical double exponential growth which behaves like exp(eeNuN′).