V. D. Rădulescu | Searcharxiv

arXiv2019

Robin problems with indefinite linear part and competition phenomena

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $λ> 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto \tilde{u}_λ$.

math.AP↗

arXiv2019

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

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 $λ<\widehatλ_{1}$ ($\widehatλ_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $λ\geq\widehatλ_{1}$ there are no positive solutions. In the superlinear case, for $λ<\widehatλ_{1}$ we have at least two positive solutions and for $λ\geq\widehatλ_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_λ$ and we investigate the properties of the map $λ\mapsto\bar{u}_λ$.

math.AP↗