Positive solutions for some indefinite nonlinear eigenvalue elliptic problems with Robin boundary conditions

Abstract

We consider a nonlinear eigenvalue problem with indefinite weight under Robin boundary condition. We prove the existence and multiplicity of positive solutions. To this end, we carry out a detailed study of some linear eigenvalues problems and we use mainly bifurcation and sub-supersolution methods.

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

“…Such positive solutions are called large positive solutions. Similar studies on large positive solutions are [19,13,1,27], where the logistic type equation −∆u = λr(x)(u − u p ) in Ω, p > 1, is considered under linear Dirichlet, Neumann or Robin boundary conditions. For λ < 0, (λ, u) is a positive solution of (4.1) if and only if (−λ, u) is a positive solution of (4.1) with r replaced by −r, by virtue of the symmetry λr(x) = (−λ)(−r(x)).…”