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

By Nikolaos S. Papageorgiou, V. Ruadulescu, Dušan D. Repovš. 2017

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

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/8d4b675c3de604b9ddfd933e48ae2bc4683841c2

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

Nonlinear nonhomogeneous singular problems (2019, background): “1 of Filippakis & Papageorgiou [9], where a similar result is proved for the upper solutions of nonlinear Dirichlet problems without singularities, driven by the p-Laplacian (see also Papageorgiou, Rădulescu & Repovš [33, 34], for Robin problems).”