An Indefinite Concave-Convex Equation Under a Neumann Boundary Condition Ii

We proceed with the investigation of the problem (Pλ) −∆u = λb(x)|u|q−2u + a(x)|u|p−2u in Ω, ∂n∂u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN (N ≥ 2), 1 < q < 2 < p, λ ∈ ℝ, and a, b ∈ Cα(Ω) with 0 < α < 1. Dealing now with the case b ≥ 0, b ≢ 0, we show the existence (and several properties) of an unbounded subcontinuum of nontrivial nonnegative solutions of (Pλ). Our approach is based on a priori bounds, a regularisation procedure, and Whyburn’s topological method. © 2017 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University.