Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis

Abstract

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators.

Double phase obstacle problems with multivalued convection and mixed boundary value conditions

“…Concerning the mathematical analysis of obstacle problems, we refer to the recent contribution of Zeng, Bai, Gasiński & Winkert [44] who applied a surjectivity theorem for multivalued mappings, Kluge's fixed point principle and tools from nonsmooth analysis to explore the existence of weak solutions for a new kind of implicit obstacle problems driven by a double phase partial differential operator and a multivalued term which is described by Clarke's generalized gradient. For further results concerning single-valued equations involving double phase operators or multivalued equations with or without double phase operators we refer to the works of Alves, Garain & Rȃdulescu [ [34,35,37], Perera & Squassina [39], Rȃdulescu [40], Vetro [42], Vetro & Vetro [43], Zhang & Rȃdulescu [47], see also the references therein. Finally, we mention the overview article of Mingione & Rȃdulescu [29] about recent developments for problems with nonstandard growth and nonuniform ellipticity.…”

Convergence analysis for double phase obstacle problems with multivalued convection term

“…Finally, papers or monographs dealing with certain types of double phase problems or multivalued problems can be found in Bahrouni-Rădulescu-Repovš [1], Bahrouni-Rădulescu-Winkert [2], [3], Carl-Le-Motreanu [10], Cencelj-Rădulescu-Repovš [11], Clarke [22], Gasiński-Papageorgiou [18], Marino-Winkert [30], Papageorgiou-Rădulescu-Repovš [32,33], Papageorgiou-Vetro-Vetro [37], Rădulescu [40], Vetro [41], Vetro-Vetro [42], Zhang-Rădulescu [45], Zeng-Bai-Gasiński-Winkert [43] and the references therein.…”

Oscillatory applications of some fourth‐order differential equations

“…We point out that advanced differential equations have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems, such as electrical power systems, control systems, networks, materials, see the book of Hale 2 . The p ‐Laplace equations have some significant applications in elasticity theory and continuum mechanics, see Aronsson and Janfalk 3 (power‐law fluids), and in general in nonlinear phenomena, see Vetro 4,5 (capillary phenomena). For some results concerning the oscillatory behavior of equations driven by a p ‐Laplace differential operator, see other studies 6‐8 for more details.…”