Existence of solutions for some weighted mean field equations in dimension N

This paper is devoted to studying the existence of solutions for the following weighted mean field equation {-div (omega(beta)(x)vertical bar del u(x)vertical bar(N-2)del u(x)) = lambda theta vertical bar u vertical bar(theta-1)e(u theta)/integral(B) (eu theta) dx, in B; (0.1) u = 0, on partial derivative B, where B = B-1(0) is the unit ball in R-N, theta = N/N-(N-1)beta, beta is an element of [0, 1) and omega(beta)(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 lambda is in some certain range. (C) 2019 Elsevier Ltd. All rights reserved.