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

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 $\lambda <{\hat{\lambda }}_1$ (${\hat{\lambda }}_1$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda \ge {\hat{\lambda }}_1$ there are no positive solutions. In the superlinear case, for $\lambda <{\hat{\lambda }}_1$ we have at least two positive solutions and for $\lambda \ge {\hat{\lambda }}_1$ 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 $\lambda \mapsto {\bar{u}}_{\lambda }$.

Cited by in corpus (12)

• Nonlinear nonhomogeneous singular problems
• Anisotropic equations with indefinite potential and competing nonlinearities
• Robin problems with indefinite linear part and competition phenomena
• Anisotropic $(p,q)$-equations with gradient dependent reaction
• Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition
• Robin problems with a general potential and a superlinear reaction
• Nonlinear, nonhomogeneous Robin problems with indefinite potential and general reaction
• The behavior of solutions of a parametric weighted (p,q)-Laplacian equation
• Parametric nonlinear resonant Robin problems
• Constant sign and nodal solutions for parametric anisotropic $(p,2)$-equations
• Nodal solutions for the Robin $p$-Laplacian plus an indefinite potential and a general reaction term
• Superlinear perturbations of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential