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

Authors:

Institution:

(1) Department of Mathematics, Yunnan Normal University, Kunming, Yunnan, 650092, P.R. China;(2) Department of Mathematics, Zhaotong Teacher’s College, Zhaotong, Yunnan, 65700, P.R. China

Abstract:

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

$\left\{ {\begin{array}{*{20}l} { - \Delta _p (u) = \lambda\, f(x,u),\quad in \quad \Omega } \\ {u = o,\quad on\quad \partial \Omega } \\ \end{array}} \right.$

((1.λ))

and

$\left\{ {\begin{array}{*{20}l} { - \Delta _p (u) + a(x)|u|^{p - 2} u = \lambda f(x,u), \quad in\quad \Omega } \\ {\frac{{\partial u}}{{\partial n}} = 0,\quad on\quad \partial \Omega } \\ \end{array}} \right.$

((2.λ))

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). This Work is supported in part by the National Natural Science Foundation of China (10561011).

Keywords:

" target="_blank"> Critical point p-Laplacian Sobolev’ s embedding Dirichlet problem Neumann problem

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