Anisotropic Robin problems with logistic reaction

Abstract

We consider Robin problems driven by the anisotropic p-Laplace operator and with a logistic reaction. Our analysis covers superdiffusive, subdiffusive and equidiffusive equations. We examine all three cases, and we prove multiplicity properties of positive solutions (superdiffusive case) and uniqueness (subdiffusive and equidiffusive cases). The equidiffusive equation is studied only in the context of isotropic operators. We explain why the more general case cannot be treated.

A multiplicity theorem for anisotropic Robin equations

“…The following estimates for γ p(•) (•) will be useful in what follows. The result can be found in the recent work of Papageorgiou-Rȃdulescu-Tang [18].…”

“…It is proved that there exists a parameter λ * > 0 such that problem (1.4) has at least two positive solutions for all λ > λ * . We also mention the works of Gasiński-Papageorgiou [11], Papageorgiou-Rȃdulescu-Tang [18] and Wang-Fan-Ge [21]. Except for [11], the above mentioned works consider parametric equations and focus on the existence and multiplicity of positive solutions.…”

On an Anisotropic Logistic Equation

“…From the isotropic literature, we know that for such problems, we have a multiplicity of positive solutions. The authors in [18] show that the same is true for p(z)-logisitic equations. They prove a multiplicity result which is global in the parameter λ > 0 (a bifurcation-type theorem), see Theorem 22 in [18].…”

“…There are no works in this direction. Only Papageorgiou-Rȃdulescu-Tang [18] considered logistic equations driven by the p(z)-Laplacian and having a Robin boundary condition. They consider the superdiffusive case (that is, the parametric term λx τ(z)−1 with p(z) < τ(z) for all z ∈ Ω).…”

Nonlinear nonhomogeneous logistic equations of superdiffusive type

“…We refer to the semilinear works of Afrouzi-Brown [4], Papageorgiou-Radulescu-Repovs [5], Radulescu-Repovs [6], and the nonlinear works of Dong [7], Gasinski-Papageorgiou [8], Papageorgiou-Radulescu-Repovs [9] (anisotropic problems), Takeuchi [10], [11]. We also mention the work of Gasinski-O'Regan-Papageorgiou [12], where the differential operator is similar to the one used here, but the aim there is to produce nodal solutions.…”