Multiplicity of Solutions to a Concave–Convex Problem

Abstract

Abstract This paper studies two related problems. A first one, where Δpu = div(|∇u|p−2∇u) stands for the p-Laplacian operator, Ω is a ball in ℝN, ν stands for the outer unit normal and λ > 0 is a parameter. Exponents are supposed to satisfy 1 < q < p < r ≤ p*, p* = N p/(N − p) if 1 < p < N, p* = ∞ otherwise. The existence of Λ > 0 is shown so that (P) does not admit positive solutions if λ > Λ, a minimal positive solution exists when 0 < λ ≤ Λ and most importantly, a further second positive solution arises if 0 < λ < Λ. Hence, extensions of results in [2], [3], [12] and [13] in the framework of (P) are provided. The second problem considered is the variant of (P): where Ω is a ball and 1 < q < p < r. Features described above are shown to be also exhibited by (Q) and more importantly, it is proved that minimal solutions to (Q) develop flat patterns in the degenerate regime p > 2. Finally, it should be stressed that some of the properties satisfied by (P) and (Q) hold true when Ω is a general smooth domain.