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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathcal S $\end{document}</tex-math></inline-formula> the solution set of the mixed boundary value problem and by <inline-formula><tex-math id="M2">\begin{document}$ \mathcal S_n $\end{document}</tex-math></inline-formula> the solution sets of the approximating problems, we establish the following convergence relation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ s $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M6">\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}</tex-math></inline-formula> stand for the weak and the strong Kuratowski upper limit of <inline-formula><tex-math id="M7">\begin{document}$ \mathcal S_n $\end{document}</tex-math></inline-formula>, respectively.</p>