On a Parametric Nonlinear Dirichlet Problem with Subdiffusive and Equidiffusive Reaction

Abstract

Abstract. We study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.

Existence and Nonexistence of Positive Solutions for Singular (p, q)-Equations with Superdiffusive Perturbation

“…More precisely, there is a critical parameter value λ * > 0 such that the problem has at least two positive solutions for all λ > λ * , the problem has at least one positive solution for λ = λ * and there are no positive solutions for λ ∈ (0, λ * ). This is in contrast to subdiffusive and equidiffusive logistic equations for which we do not have multiplicity of positive solutions, see Papageorgiou-Winkert [19].…”

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.…”

On a Class of Parametric (p, 2)-equations

“…(see Aizicovici, Papageorgiou and Staicu [1] (proof of Proposition 22)). The functionalφ λ is coercive, hence it satisfies the C-condition (see [30]). This fact and (75) permit the use of Theorem 2 (the mountain pass theorem).…”