Binding, Paul A.Drabek, PavelHuang, Yin Xi2018-08-302018-08-301997-01-30Binding, 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. 1-11.1072-6691https://hdl.handle.net/10877/7658We 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 ℝⁿ, b changes sign, 1 < p < N, 1 < γ < Np/ (N − p) and γ ≠ p. We prove that (i) if ∫_Ω 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.Text11 pages1 file (.pdf)enAttribution 4.0 Internationalp-Laplacianpositive solutionsNeumann boundary value problemsArticle