This paper studies the existence of nontrivial solutions to the following class of Schrödinger equations: {−div(w(x)∇u)=f(x,u),x∈B1(0),u=0,x∈∂B1(0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\operatorname{div}(w(x)\ abla u) = f(x,u),&\\ x \\in B_{1}(0), \\\\ u = 0,&\\ x \\in \\partial B_{1}(0), \\end{cases} $$\\end{document} where w(x)= (ln (1/|x|) )^{beta} for some beta in [0,1), the nonlinearity f(x,s) behaves like {exp} (|s|^{frac{2}{1-beta}+h(|x|)} ), and h is a continuous radial function such that h(r) can be unbounded as r tends to 1. Our approach is based on a new Trudinger–Moser-type inequality for weighted Sobolev spaces and variational methods.