Existence results for double phase implicit obstacle problems involving multivalued operators
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Calculus of Variations
Existence results for double phase implicit obstacle problems involving multivalued operators Shengda Zeng1,2 · Yunru Bai2 · Leszek Gasinski ´ 3 · Patrick Winkert4 Received: 16 March 2020 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution. Mathematics Subject Classification 35J20 · 35J25 · 35J60
1 Introduction Given a bounded domain in R N , N ≥ 2, with Lipschitz boundary ∂, we study a double phase implicit obstacle problem with a multivalued operator given in the form
Communicated by P. Rabinowitz.
B
Shengda Zeng [email protected] Yunru Bai [email protected] Leszek Gasi´nski [email protected] Patrick Winkert [email protected]
1
Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, People’s Republic of China
2
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30-348 Kraków, Poland
3
Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Kraków, Poland
4
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany 0123456789().: V,-vol
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−div |∇u| p−2 ∇u + μ(x)|∇u|q−2 ∇u + ∂ j(x, u) f (x) u=0
in , on ∂,
(1.1)
T (u) ≤ U (u), where 1 < p < q < N , μ : → [0, ∞) and T , U : W01,H () → R are given functions satisfying appropriate conditions (see Sect. 2). Here W01,H () is a subspace of the Sobolev– Musielak–Orlicz space W 1,H () and j : × R → R is supposed to be locally Lipschitz with respect to the second variable. In this paper we prove the existence of at least one weak solution (see Definition (3.4)) of problem (1.1) by applying a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis. In general, problem (1.1) combines several interesting phenomena like a double phase operator along with a multivalued mapping in form of Clarke’s generalized gradient and an implicit obstacle given by the functions T : W01,H () → R and U : W01,H () → (0, +∞), see H(T ) and H(U ) in Sect. 3 for the precise conditions. Indeed, a solution u ∈ W01,H () of (1.1) has to belong to K (u) which is 1,H
the image of the multivalued map K : W01,H () → 2W0 () defined by K (u) := v ∈ W01,H () | T (v) − U (u) ≤ 0 .
To the best of our knowledge, this is the first work which combines a double phase phenomena along with Clarke’s generalized gradient and an implicit obstacle. A main tool in our treatment will be a surjectivity result of Le [24] for multivalued mappings generated by
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