Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type. (January 2015) - Sabina de Lis, José C.; Segura de León, Sergio
Abstract: Problem (P) { − Δ p u + | u | p − 2 u = | u | r − 1 u x ∈ Ω | ∇ u | p − 2 ∂ u ∂ ν = λ | u | s − 1 u x ∈ ∂ Ω, where Ω ⊂ R N is a bounded smooth domain, ν is the unit outward normal at ∂ Ω, Δ p is the p -Laplacian operator and λ > 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 1 < s < p < r, a minimal positive solution exists if 0 < λ ≤ Λ, for a certain finite Λ, while no positive solutions exist in the complementary range λ > Λ . Furthermore, in the radially symmetric case a second positive solution exists for λ varying in the same full range ( 0, Λ ) provided r < p ∗ . Our main achievement in this work just asserts that such global multiplicity feature holds true when Ω is a general domain. To show such result the well-known Brezis–Nirenberg variational result in Brezis and Nirenberg (1993) must be extended to the framework of(P) . This is the second main contribution in the present work.