Multiple solutions to a pure supercritical problem for the p-Laplacian

Under some symmetry assumptions, we establish existence and multiplicity of positive and sign changing solutions to the quasilinear problem -Δpv=vq-2vinΩ,v=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta _{p}v=\left| v\right| ^{q-2}v\quad \text {in }\Omega ,\quad v=0\quad \text {on }\partial \Omega , \end{aligned}$$\end{document}in some bounded smooth domains Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{N}$$\end{document}, for 1<p<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<N$$\end{document} and some supercritical exponents q>p∗:=NpN-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>p^{*}:=\frac{Np}{N-p}$$\end{document}. Related to this question is the existence of solutions to the anisotropic critical problem -div(a(x)∇up-2∇u)=c(x)up∗-2uinΩ,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}$$\begin{aligned} -\text {div}(a(x)\left| \nabla u\right| ^{p-2}\nabla u)=c(x)\left| u\right| ^{p^{*}-2}u\quad \text {in } \Omega ,\quad u=0\quad \text {on } \partial \Omega , \end{aligned}$$\end{document}where a and c are positive continuous functions on Ω¯.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega } .$$\end{document} We study the lack of compactness for this problem in a symmetric setting, and establish existence and multiplicity of positive and sign changing solutions by means of a suitable variational principle.