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Positive solutions for Dirichlet problems of singular quasilinear elliptic equations via variational methods
Author(s) -
Agarwal Ravi P.,
Lü Haishen,
O'Regan Donal
Publication year - 2004
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015618
Subject(s) - mathematics , bounded function , multiplicity (mathematics) , dirichlet problem , domain (mathematical analysis) , mathematical analysis , variational principle , maximum principle , pure mathematics , boundary value problem , mathematical optimization , optimal control
This paper studies the existence and multiplicity of positive solutions of the following problem: *{− Δ p u =u βδ γ ( x ) + λ a ( x )u α,in Ω ,u > 0 ,in Ω ,u = 0 ,on ∂ Ω ,where Ω⊂ R N ( N ≥3) is a smooth bounded domain,Δ p u = d i v (| ∇ u |p − 2 ∇ u ) , 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 ∈ L p 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.