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Positive solutions for Dirichlet problems of singular quasilinear elliptic equations via variational methods

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  26 February 2010

Ravi P. Agarwal ,
Haishen Lü  and
Donal O'Regan
Ravi P. Agarwal
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, U.S.A.
Haishen Lü
Affiliation:
Department of Applied Mathematics, Hohai University, Nanjing, 210098, China.
Donal O'Regan
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland.
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Abstract

This paper studies the existence and multiplicity of positive solutions of the following problem:

where Ω⊂RN(N≥3) is a smooth bounded domain, , 1 < p < N, and 0 < α < 1, p - 1 < β < p* - 1 (p* = Np/(N - p)) and 0 < γ < N + ((β + 1)(p - N)/p) are three constants. Also δ(x) = dist(x, ∂Ω), a ∈ Lp and λ < 0 is a real parameter. By using the direct method of the calculus of variations, Ekeland's Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if λ is small enough.

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 187 - 202
Copyright
Copyright © University College London 2004

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