Abstract
We prove existence and comparison results for multi-valued variational inequalities in a bounded domain Ω of the form (Formula presented.) where A:W1,H(Ω)→W1,H(Ω)∗ given by (Formula presented.) for u∈W1,H(Ω), is the double phase operator with variable exponents and W1,H(Ω) is the associated Musielak–Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub–super-solution for the multi-valued variational inequality in the space W1, H(Ω) based on appropriately defined sub- and super-solutions, which yields the existence of solutions within an ordered interval of sub–super-solution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset K of W1, H(Ω) satisfies a lattice condition. As an application of the sub–super-solution method we are able to show that a class of generalized variational–hemivariational inequalities with a leading double phase operator are included as a special case of the multi-valued variational inequality considered here. Based on a fixed-point argument, we also study the case when the corresponding Nemytskij operators F,FΓ need not be continuous. At the end, we give an example of the construction of sub- and super solutions related to the problem above.
Recommended Citation
S. Carl et al., "Multi-valued Variational Inequalities For Variable Exponent Double Phase Problems: Comparison And Extremality Results," Journal of Elliptic and Parabolic Equations, Springer, Jan 2025.
The definitive version is available at https://doi.org/10.1007/s41808-025-00319-6
Department(s)
Mathematics and Statistics
Publication Status
Open Access
Keywords and Phrases
Comparison results; Discontinuous problems; Extremality results; Multi-valued variational inequalities; Musielak–Orlicz Sobolev space; Obstacle problem; Sub- and super-solution; Variable exponent double phase operator
International Standard Serial Number (ISSN)
2296-9039; 2296-9020
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2025 Springer, All rights reserved.
Creative Commons Licensing
This work is licensed under a Creative Commons Attribution 4.0 License.
Publication Date
01 Jan 2025
