Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

Nikolaos S Papageorgiou,Dušan D Repovš,Vicenţiu D Rădulescu

doi:10.3934/dcds.2017111

Abstract

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.

Highlights

Let Ω ⊆ RN be a bounded domain with a C 2 -boundary ∂Ω
We look for positive solutions and consider two cases: a sublinear perturbation f (z, ·) and a superlinear
We mention that the standard eigenvalue problems for the Robin Laplacian have recently been studied by D’Agui, Marano & Papageorgiou [4] and by Papageorgiou & Rădulescu [11]

Summary

Introduction

In this paper we study the following semilinear parametric Robin problem with an indefinite and unbounded potential ξ(z):. We can think of (Pλ ) as a perturbation of the standard eigenvalue problem for the differential operator u 7→ −∆u + ξ(z)u with Robin boundary condition. We look for positive solutions and consider two cases: a sublinear perturbation f (z, ·) and a superlinear. Indefinite and unbounded potential, Robin eigenvalue problem, sublinear perturbation, superlinear perturbation, maximum principle, positive solution, minimal positive solution. For both cases we determine the dependence of the set of positive solutions as the parameter λ ∈ R varies. We mention that the standard eigenvalue problems for the Robin Laplacian have recently been studied by D’Agui, Marano & Papageorgiou [4] and by Papageorgiou & Rădulescu [11]. For the convenience of the reader, we recall the main mathematical tools which will be used in the sequel

Mathematical background

The sublinear case

The superlinear case