Asymptotic behavior of positive solutions of a Dirichlet problem involving combined nonlinearities

We study the behavior of positive solutions of the following Dirichlet problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in} \enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array} \right. $$\end{document}when s → p−. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p >1 , s\,{\in}\,]1,p]}$$\end{document} and q > p with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q\leq\frac{Np}{N-p}}$$\end{document} if N > p.