Inverse problems for anisotropic obstacle problems with multivalued convection and unbalanced growth

The prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic \begin{document}$ p $\end{document}-Laplacian and a weighted anisotropic \begin{document}$ q $\end{document}-Laplacian (called the weighted anisotropic \begin{document}$ (p,q) $\end{document}-Laplacian), a multivalued reaction term depending on the gradient, two multivalued boundary conditions and an obstacle constraint. We, first, employ the theory of nonsmooth analysis and a surjectivity theorem for pseudomonotone operators to prove the existence of a nontrivial solution of the anisotropic elliptic obstacle problem, which relies on the first eigenvalue of the Steklov eigenvalue problem for the \begin{document}$ p\_$\end{document}-Laplacian. Then, we introduce the parameter-to-solution map for the anisotropic elliptic obstacle problem, and establish a critical convergence result of the Kuratowski type to parameter-to-solution map. Finally, a general framework is proposed to examine the solvability of the nonlinear inverse problem.