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

  • *Corresponding author: Vicenţiu D. Rădulescu

    *Corresponding author: Vicenţiu D. Rădulescu 
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  • In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $ \mathcal S $ the solution set of the mixed boundary value problem and by $ \mathcal S_n $ the solution sets of the approximating problems, we establish the following convergence relation

    $ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} $

    where $ w $-$ \limsup_{n\to\infty}\mathcal S_n $ and $ s $-$ \limsup_{n\to\infty}\mathcal S_n $ stand for the weak and the strong Kuratowski upper limit of $ \mathcal S_n $, respectively.

    Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60.


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