CPAA
Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term
Nikolaos S. Papageorgiou Vicenţiu D. Rǎdulescu Dušan D. Repovš

We consider a nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in $C^1(\overline{Ω})$.

keywords: Robin p-Laplacian indefinite potential nodal solutions truncation techniques comparison principle
CPAA
Robin problems with indefinite linear part and competition phenomena
Nikolaos S. Papageorgiou Vicenšiu D. Rădulescu Dušan D. Repovš

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 $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.

keywords: Indefinite potential Robin boundary condition strong maximum principle truncation competing nonlinear positive solutions regularity theory minimal positive solution
DCDS
Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. Repovš

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<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{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}$.

keywords: Indefinite and unbounded potential Robin eigenvalue problem sublinear perturbation superlinear perturbation maximum principle positive solution minimal positive solution

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