Regularity of the Extremal Solution of Some Nonlinear Elliptic Problems Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Regularity of the Extremal Solution of Some Nonlinear Elliptic Problems Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

Authors:	Manel Sanchón

Institution:	(1) Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

Abstract:	We consider the equation $- {\text{div}}{\left( {{\left\| {\nabla u} \right\|}^{{p - 2}} \nabla u} \right)} = \lambda f{\left( u \right)}$ on a smooth bounded domain of $\mathbb{R}^{N}$ with zero Dirichlet boundary conditions where p ≥ 2, λ > 0 and f satisfies typical assumptions in the subject of extremal solutions. We prove that, for such general nonlinearities f, the extremal solution u ^* belongs to L ^∞(Ω) if N < p + p/(p − 1) and $u^{*} \in W^{{1,p}}_{0} {\left( \Omega \right)}$ if N < p(1 + p/(p − 1)). This work was partially supported by MCyT BMF 2002-04613-CO3-02.

Keywords:	p-Laplacian Extremal solution Regularity Semi– stability
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