We consider the following elliptic problem: \t\t\t{−div(|∇u|p−2∇u|y|ap)=|u|q−2u|y|bq+f(x)in Ω,u=0on ∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\operatorname{div} ( \\frac{ \\vert \\nabla u \\vert ^{p-2} \\nabla u}{ \\vert y \\vert ^{ap}} ) = \\frac { \\vert u \\vert ^{q-2} u}{ \\vert y \\vert ^{bq}} + f(x) & \\mbox{in } \\Omega,\\\\ u = 0 & \\mbox{on } \\partial\\Omega, \\end{cases} $$\\end{document} in an unbounded cylindrical domain \t\t\tΩ:={(y,z)∈Rm+1×RN−m−1;0<A<|y|<B<∞},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Omega:= \\bigl\\{ (y,z)\\in \\mathbb{R}^{m+1}\\times\\mathbb{R}^{N-m-1} ; 0< A< \\vert y \\vert < B < \\infty \\bigr\\} , $$\\end{document} where 1leq m< N-p, q=q(a,b):=frac{Np}{N-p(a+1-b)}, p>1 and A,Binmathbb{R}_{+}. Let p^{*}_{N,m}:=frac {p(N-m)}{N-m-p}. We show that p^{*}_{N,m} is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Maz’ja’s inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev (p=2, a=0 mbox{ and } b=0) and Hardy (p=2, a=0 mbox{ and } b=1). Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case fequiv0 and at least two solutions in the case f notequiv0, if p< q< p^{*}_{N,m}.