Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

Abstract

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical "convex-concave" problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter λ>0 varies. We also show the existence of a smallest positive solution ūλ and investigate the properties of the map λ→ūλ. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Document Type

Article

DOI

https://doi.org/10.1142/s021919971550090x

Keywords

nonlinear boundary condition, competing nonlinearity, bifurcation-type theorem, extremal constant sign solutions, nodal solutions, indefinite and unbounded potential

Publication Date

2017

Journal Title

Communications in Contemporary Mathematics

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