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,cdot ) 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 lambda in {mathbb {R}} which we specify exactly in terms of principal eigenvalue of the differential operator.
Highlights
Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω
We look for positive solutions and we consider two distinct cases depending on the growth of the perturbation f (z, ·) near +∞:
In the first case (( p −1)-sublinear perturbation), we show that for all λ ≥ λ1, problem (Pλ) has no positive solution, while for λ < λ1, problem (Pλ) has at least one positive solution
Summary
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). Problem (Pλ) can be viewed as a perturbation of the usual eigenvalue problem for the Robin p-Laplacian plus an indefinite potential. Eigenvalue problems for the p-Laplacian plus an indefinite potential were studied by Papageorgiou–Radulescu [18] (semilinear problems (that is, p = 2) with Robin boundary condition) and by Mugnai–Papageorgiou [16] (nonlinear problems with Neumann boundary condition (that is, β ≡ 0)). Both works deal with nonparametric problems and prove existence and multiplicity results under resonance conditions. In [11] the authors treat superdiffusive logistic equation with Robin boundary condition, while in [10,12], they deal with equations driven by a nonhomogeneous differential operator
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