Inverse problems for a class of elliptic obstacle problems involving multivalued convection term and weighted $(p,q)$-Laplacian

Abstract

In this paper, we study the inverse problem of estimating discontinuous parameters and boundary data in a nonlinear elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is given as a sum of a weighted p-Laplacian and a weighted q-Laplacian (called the weighted -Laplacian), a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. Under general hypotheses on the data, we prove the existence of a nontrivial solution to the elliptic obstacle problem, which depends on the first eigenvalue of the Steklov eigenvalue problem for the p-Laplacian. We also establish the boundedness and the closedness of the solution set to the problem. For the parameter-to-solution map, we give a convergence result of the Kuratowski type and prove the solvability of the inverse problem.