Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

Abstract

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ * > 0 such that for λ < λ * , the problem has no positive solution; for λ = λ * , it has at least one positive solution; and for λ > λ * , it has at least two positive solutions.

Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential

“…Their work was extended recently to problems with a (p − 1)−superlinear reaction term by Fragnelli -Mugnai -Papageorgiou [10] (see also Mugnai -Papageorgiou [21]). We mention also the relevant works of Brock -Iturriaga -Ubilla [5] (nonlinear parametric Dirichlet problems with ξ ≡ 0), Cardinali -Papageorgiou -Rubbioni [6] (nonlinear parametric Neumann problems with ξ ≡ 0 and a superdiffusive reaction term), Gasinski -Papageorgiou [11] (nonlinear parametric Dirichlet problems with ξ ≡ 0 and a logistic reaction term), Papageorgiou -Radulescu [24] (nonlinear parametric Robin problems with ξ ≡ 0, the parameter λ > 0 multiplying the reaction term and the latter satisfying certain monotonicity properties) and Takeuchi [28], [29] (semilinear superdiffusive logistic equations driven by the Dirichlet Laplacian with zero potential). Finally, we recall also the work of Mugnai -Papageorgiou [22] on logistic equations on R N driven by the Dirichlet p−Laplacian with zero potential.…”

Bifurcation near infinity for the Robin p-Laplacian

“…Such bifurcation type results, were proved by Brock et al [3], Filippakis et al [8], Gasinski and Papageorgiou [10], Rȃdulescu and Repovs [17], Takeuchi [19,20] (semilinear or nonlinear Dirichlet problems) and by Cardinali et al [4], Papageorgiou and Rȃdulescu [15] (for nonlinear Neumann problems). All the aforementioned results impose more restrictive conditions on the reaction f (z, ·).…”

Bifurcation for positive solutions of nonlinear diffusive logistic equations in R^N with indefinite weight

“…Problem (P λ ) settled in a bounded region with Neumann boundary condition was studied by Cantrell-Cosner-Hutson [4] and Umezu [25], and extensions to the p-Laplacian Dirichlet case can be found in the works of Dong [7], Garcia Melian-Sabina de Lis [15], Guedda-Veron [17], Kamin-Veron [19] and Papageorgiou-Papalini [21]. A related Neumann problem can be found also in Cardinali-Papageorgiou-Rubbioni [5]. Diffusive logistic equations in the whole of R N were studied by Afrouzi-Brown [2], who dealt with the semilinear equation (i.e.…”