Perturbed eigenvalue problems for the Robin p -Laplacian plus an indefinite potential
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Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential Calogero Vetro1 Received: 17 September 2019 / Revised: 17 September 2019 / Accepted: 18 October 2020 © The Author(s) 2020
Abstract We consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation f (z, ·) is ( p − 1)sublinear and then the case where it is ( p − 1)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter λ ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator. Keywords Positive solutions · Sublinear and superlinear perturbation · Nonlinear Picone’s identity · Nonlinear maximum principle · Nonlinear regularity · Indefinite potential · Minimal positive solution · Uniqueness Mathematics Subject Classification Primary: 35J20; Secondary: 35J60
1 Introduction Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω. In this paper we study the following nonlinear parametric Robin problem ⎧ ⎨−Δ p u(z) + ξ(z)|u(z)| p−2 u(z) = λ|u(z)| p−2 u(z) + f (z, u(z)) ∂u ⎩ + β(z)|u| p−2 u = 0 ∂n p
in Ω, on ∂Ω, u ≥ 0, λ ∈ R.
(Pλ )
In this problem Δ p denotes the p-Laplace differential operator defined by Δ p u = div (|∇u| p−2 ∇u)
B 1
for all u ∈ W 1, p (Ω) (1 < p < +∞).
Calogero Vetro [email protected] Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy 0123456789().: V,-vol
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C. Vetro
Also ξ(·) ∈ L ∞ (Ω) is an indefinite (that is, sign changing) potential function, λ ∈ R is a parameter and f (z, x) is a Carathéodory perturbation function (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous). ∂u denotes the generalized normal derivative defined by In the boundary condition ∂n p ∂u ∂u = |∇u| p−2 (∇u, n)R N = |∇u| p−2 for all u ∈ W 1, p (Ω), ∂n p ∂n with n(·) being the outward unit normal on ∂Ω. This kind of generalized normal derivative is dictated by the nonlinear Green’s identity (see, for example, Gasi´nski– Papageorgiou [8] (p. 211)). The boundary weight term β ∈ C 0,α (∂Ω) (0 < α < 1) and β(z) ≥ 0 for all z ∈ ∂Ω. Problem (Pλ ) can be viewed as a perturbation of the usual eigenvalue problem for the Robin p-Laplacian plus an indefinite potential. We look for positive solutions and we consider two distinct cases depending on the growth of the perturbation f (z, ·) near +∞: • f (z, ·) is ( p − 1)-sublinear. • f (z, ·) is ( p − 1)-superlinear. Let λ1 ∈ R be the principal eigenvalue of the differential operator u → −Δ p u + ξ(z)|u| p−2 u with Robin boundary condition. In the first case (( p −1)-sublinear perturbation), we show that for all λ ≥ λ1 , problem (Pλ ) has no positive solution, while for λ 0. In the second case (( p −1)-superlinear perturbation), the s
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