On Neumann Boundary Value Problems for Some Quasilinear Elliptic Equations
Date
1997-01-30
Authors
Binding, Paul A.
Drabek, Pavel
Huang, Yin Xi
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We study the role played by the indefinite weight function a(x) on the existence of positive solutions to the problem
{−div (|∇u|p−2</sup> ∇u) = λα(x) |u|p−2 u + b(x)|u|γ−2 u, x ∈ Ω, ∂u / ∂n = 0, x ∈ ∂Ω ,
where Ω is a smooth bounded domain in ℝ<sup>n</sup>, b changes sign, 1 < p < N, 1 < γ < Np/ (N − p) and γ ≠ p. We prove that (i) if ∫<sub>Ω</sub> α(x) dx ≠ 0 and b satisfies another integral condition, then there exists some λ* such that λ* ∫Ω α(x) dx < 0 and, for λ strictly between 0 and λ*, the problem has a positive solution and (ii) if ∫Ω α(x) dx = 0, then the problem has a positive solution for small λ provided that ∫Ω b(x) dx < 0.
Description
Keywords
p-Laplacian, positive solutions, Neumann boundary value problems
Citation
Binding, P. A., Drabek, P., & Huang, Y. X. (1997). On Neumann boundary value problems for some quasilinear elliptic equations. <i>Electronic Journal of Differential Equations, 1997</i>(05), pp. 1-11.
Rights
Attribution 4.0 International