A bifurcation-type theorem for singular nonlinear elliptic equations

Abstract

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian and exhibiting the combined effects of singular and superlinear terms. Using variational methods combined with truncation and comparison techniques, we prove a bifurcation-type theorem. More precisely, we show that there exists a critical parameter value λ * > 0 s.t. for all λ ∈ (0, λ *) (λ being the parameter) the problem has at least two positive smooth solutions, for λ = λ * the problem has at least one positive smooth solution and for λ > λ * the positive solutions disappear.

Nonlinear singular problems with indefinite potential term

“…Using Proposition 3 of Papageorgiou and Smyrlis [17], we see thatσ λ ∈ C 1 (W 1, p 0 ( )). Also, from (26), Corollary 3 and since β > ||ξ || ∞ , we see thatσ λ (•) is coercive.…”

“…This work continues the recent paper by Papageorgiou et al [16], where ξ ≡ 0 and in the reaction the parametric term is the singular one. It is also related to the works of Papageorgiou and Smyrlis [17] and Papageorgiou and Winkert [19], where the differential operator is the p-Laplacian, ξ ≡ 0 and no concave terms are allowed. Singular p-Laplacian equations with no potential term and reactions of special form were considered by Chu et al [2], Giacomoni et al [5], Li and Gao [10], Mohammed [12], Perera and Zhang [20], and Papageorgiou et al [14].…”

“…The following strong comparison principle can be found in Papageorgiou et al [16,Proposition 6] (see also Papageorgiou and Smyrlis [17,Proposition 4]).…”

Positive solutions for nonlinear parametric singular Dirichlet problems

“…The existence of a unique solutionũ λ ∈ int C + follows from Proposition 5 of Papageorgiou & Smyrlis [18].…”

“…The next strong comparison principle can be found in Papageorgiou & Smyrlis [18,Proposition 4] (see also Giacomoni,Schindler & Takač [7,Theorem 2.3]).…”