On a Robin (p,q)-equation with a logistic reaction

Abstract

We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a p-Laplacian and of a q-Laplacian ((p, q)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter λ > 0 varies. Also, we show that for every admissible parameter λ > 0, the problem admits a smallest positive solution.

Anisotropic Robin problems with logistic reaction

“…In the past, nonlinear logistic equations were investigated only in the framework of equations with differential operators which have constant exponents. We mention the works of Cardinali et al [4], Dong and Chen [7], Filippakis et al [11], Papageorgiou et al [19], Papageorgiou et al [23], Takeuchi [31,32] (superdiffusive problems), El Manouni et al [8], Winkert [34] (nonhomogeneous Neumann problems), and Ambrosetti and Lupo [2], Ambrosetti and Mancini [3], Kamin and Veron [15], D'Aguì et al [5], Papageorgiou and Papalini [17], Papageorgiou and Scapellato [22], Papageorgiou and Winkert [24], Papageorgiou and Zhang [25], Rȃdulescu and Repovš [26], Struwe [28,29] (subdiffusive and equidiffusive equations). Moreover, of the above works only the one by Papageorgiou et al [23], considers Robin boundary value problems.…”

Multiple solutions for parametric double phase Dirichlet problems

“…x 0 f (z, s)ds. According to hypothesis H 1 (iii) given ε > 0 we can find δ 0 ∈ (0, 1) such that (14) F (z, x) ≥ − ε q |x| q for a.a. z ∈ Ω, all |x| ≤ δ 0 .…”

“…The study of equations in which the energy density changes its ellipticity and growth properties according to the point of the domain, has been revived more recently by Mingione and coworkers, who in a series of remarkable papers produced local regularity results (see [1,2,5,6]). However, no global regularity results are yet available for two phase elliptic problems and this fact does not allow the application of many of the tools and techniques used in the study of (p, q)-equations (see, for example, Papageorgiou-Vetro-Vetro [14]). Neverthless, using variational methods and suitable truncation and comparison techniques, we show that if λ 2 (q) is the second eigenvalue of (−∆ q , W 1,q 0 (Ω)) and λ > λ 2 (q), then problem (P λ ) admits at least three nontrivial solutions.…”

“…We require that f (z, x) is a Carathéodory function (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous) and f (z, •) exhibits (p − 1)-superlinear growth near ±∞ and (q − 1)sublinear growth near 0. So, the reaction of problem (P λ ) is of logistic type and more precisely of the equidiffusive kind (see [14]).…”

A multiplicity theorem for parametric superlinear (p,q)-equations

“…Parametric (p, q)-equations were studied primarily in the context of Dirichlet problems, using different conditions of the reaction. We mention the works of Benouhiba-Belyacine [4], Bhattacharya-Emamizadeh-Farjudian [5], Bobkov-Tanaka [6], Papageorgiou-Rǎdulescu [15], Papageorgiou-Rǎdulescu-Repovš [19,20], Papageorgiou-Vetro-Vetro [22], Papageorgiou-Zhang [25,26], Rǎdulescu [28], Tanaka [29].…”