Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type

Problem equation(P){−Δpu+|u|p−2u=|u|r−1ux∈Ω|∇u|p−2∂u∂ν=λ|u|s−1ux∈∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain, νν is the unit outward normal at ∂Ω∂Ω, ΔpΔp is the pp-Laplacian operator and λ>0λ>0 is a parameter, was studied in Sabina de Lis (2011) and Sabina de Lis and Segura de León (in press). Among other features, it was shown there that when exponents lie in the regime 1Λλ>Λ. Furthermore, in the radially symmetric case a second positive solution exists for λλ varying in the same full range (0,Λ)(0,Λ) provided r