Existence and Multiplicity Results for an Elliptic Problem Involving Cylindrical Weights and a Homogeneous Term μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}

We consider the following elliptic problem -div∇up-2∇uyap=μup-2uyp(a+1)+h(x)uq-2uybq+f(x,u)inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lll} -{\text {div}}\left( \dfrac{\left| \nabla u\right| ^{p-2} \nabla u}{\left| y\right| ^{ap}}\right) = \mu \dfrac{\left| u\right| ^{p-2} u}{\left| y\right| ^{p(a+1)}}+ h(x) \dfrac{\left| u\right| ^{q-2} u}{\left| y\right| ^{bq}} + f(x,u) &{}&{} \text{ in } \ \Omega , \\ u = 0 &{}&{} \text{ on } \ \partial \Omega ,\\ \end{array} \right. \end{aligned}$$\end{document}in an unbounded cylindrical domain Ω:={(y,z)∈Rm+1×RN-m-1;0<A<y<B<∞},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega :=\{ (y,z)\in {\mathbb {R}}^{m+1}\times {\mathbb {R}}^{N-m-1} \ ; \ 0<A<\left| y\right|<B <\infty \}, \end{aligned}$$\end{document}where A,B∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A,B\in {\mathbb {R}}_+$$\end{document}, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, 1≤m<N-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le m<N-p$$\end{document}, q:=NpN-p(a+1-b),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q:=\dfrac{Np}{N-p (a+1-b)},$$\end{document}0≤μ<μ¯:=m+1-p(a+1)pp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \mu < {\overline{\mu }}:=\left( \dfrac{m+1-p(a+1)}{p}\right) ^p $$\end{document}, h∈LNq(Ω)∩L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in L^{\frac{N}{q}}(\Omega )\cap L^{\infty }(\Omega )$$\end{document} is a positive function and f:Ω×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: \Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} is a Carathéodory function with growth at infinity. Using the Krasnoselski’s genus and applying Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_2$$\end{document} version of the Mountain Pass Theorem, we prove, under certain assumptions about f, that the above problem has infinite invariant solutions.