Existence of solutions for some weighted mean field equations in dimension [formula omitted

Abstract

This paper is devoted to studying the existence of solutions for the following weighted mean field equation (0.1)−div(ωβ(x)|∇u(x)|N−2∇u(x))=λθ|u|θ−1euθ∫Beuθdx,inB;u=0,on∂B,where B=B1(0) is the unit ball in RN, θ=NN−(N−1)β, β∈[0,1) and ωβ(x) is of logarithmic type function. By showing the logarithmic Trudinger–Moser inequalities in the weighted Sobolev spaces, we establish the existence of solutions for problem (0.1) when λ is in some certain range.