A remark on the existence and multiplicity result for a nonlinear elliptic problem involving the p-Laplacian

In this work, motivated by Wu (J Math Anal Appl 318:253–270, 2006), and using recent ideas from Brown and Wu (J Math Anal Appl 337:1326–1336, 2008), we prove the existence of nontrivial nonnegative solutions to the following nonlinear elliptic problem: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll} -\Delta_{p}u+m(x)\,u^{p-1}=\lambda \,a(x)\, u^{\alpha-1}+b(x)\,u^{\beta-1}, & x \in \Omega,\\ u=0, & x\in\partial\Omega. \end{array}\right.$$\end{document}Here Δp denotes the p-Laplacian operator defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_{p}z=div\,(|\nabla z|^{p-2}\nabla z),\, p > 2,\Omega\subset \mathbb{R}^N}$$\end{document} is a bounded domain with smooth boundary, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2 < \beta < p < \alpha < p* (p*=\frac{pN}{N-p}\, {\rm if}\, N > p,\, p*=\infty\, {\rm if}\, N\leq p),\, \lambda \in \mathbb{R} \setminus \{0\}}$$\end{document} , the weight m(x) is a bounded function with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\| m\|_{\infty} > 0}$$\end{document} and a(x), b(x) are continuous functions which change sign in \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}.