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.
Highlights
Given a bounded domain in RN, N ≥ 2, with Lipschitz boundary ∂, we study a double phase implicit obstacle problem with a multivalued operator given in the form
Of problem (1.1) by applying a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis
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
Summary
Given a bounded domain in RN , N ≥ 2, with Lipschitz boundary ∂ , we study a double phase implicit obstacle problem with a multivalued operator given in the form. 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. 2 we recall the definition of the used function spaces, some embedding results and we state the surjectivity results of Le [24] for multivalued mappings as well as Kluge’s fixed point theorem. Taking these results into account we are able to prove our main result which says that the solution set of (1.1) is nonempty, bounded and weakly closed in W01,H( ), see Theorem 3.5
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