On Neumann Boundary Value Problems for Some Quasilinear Elliptic Equations




Binding, Paul A.
Drabek, Pavel
Huang, Yin Xi

Journal Title

Journal ISSN

Volume Title


Southwest Texas State University, Department of Mathematics


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.



p-Laplacian, positive solutions, Neumann boundary value problems


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.


Attribution 4.0 International

Rights Holder

Rights License