On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian (1.λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\begin{array}{*{20}l} { - \Delta _p (u) = \lambda\, f(x,u),\quad in \quad \Omega } \\ {u = o,\quad on\quad \partial \Omega } \\ \end{array}} \right.$$\end{document} and (2.λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\begin{array}{*{20}l} { - \Delta _p (u) + a(x)|u|^{p - 2} u = \lambda f(x,u), \quad in\quad \Omega } \\ {\frac{{\partial u}}{{\partial n}} = 0,\quad on\quad \partial \Omega } \\ \end{array}} \right. $$\end{document} are studied and some new multiplicity results of solutions for systems (1.λ) and (2.λ) are obtained. Moreover, by using the KKM principle we give also two new existence results of solutions for systems (1.1) and (2.1).