Positive solutions for superdiffusive mixed problems

Abstract

We study a semilinear parametric elliptic equation with superdiffusive reaction and mixed boundary conditions. Using variational methods, together with suitable truncation techniques, we prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions

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.…”

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

“…Parametric superdiffusive logistic equations with no singular term present, were investigated by Afrouzi-Brown [1] (for semilinear Dirichlet problems), Takeuchi [23,24] (for nonlinear Dirichlet problems driven by the p-Laplacian), Gasiński-O'Regan-Papageorgiou [3] (for nonlinear Dirichlet problems driven by a nonhomogeneous differential operator), Cardinali-Papageorgiou-Rubbioni [2], Gasiński-Papageorgiou [7] (both dealing with nonlinear problems driven by the p-Laplacian) and Papageorgiou-Rȃdulescu-Repovš [16] (for semilinear mixed problems). These works reveal that the superdiffusive logistic equations exhbit a kind of "bifurcation" for large values of the parameter λ > 0.…”

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

“…The differential operator of problem (P λ ) is nonhomogeneous and this is a source of difficulties in the analysis of problem (P λ ). Many of the arguments in the study of superdiffusive logistic equations driven by the Laplacian or p-Laplacian depend heavily on the homogeneity of the operator (see, for example [8,18,19,22,24]). So, in the present setting they have to be modified.…”

“…Subdiffusive and equidiffusive equations were examined in Ambrosetti-Lupo [2], Ambrosetti-Mancini [3], Marano-Papageorgiou [16], Rǎdulescu-Repovš [27], Struwe [28,29] (all dealing with semilinear Dirichlet equations driven by the Laplacian) and also Kamin-Veron [14], Marano-Papageorgiou [16], Papageorgiou-Papalini [18,19], Papageorgiou-Winkert [22] (nonlinear equations driven by the p-Laplacian). The superdiffusive case was investigated by Cardinali-Papageorgiou-Rubbioni [4], Dong-Chen [7], Filippakis-O'Regan-Papageorgiou [8], Papageorgiou-Rădulescu-Repovs [24], Takeuchi [30,31] (nonlinear equations driven by the Laplacian and p-Laplacian).…”