Abstract
By using the monotonicity, hemicontinuity, and pseudomonotonicity of the considered integral functionals, we studied the well-posedness of some variational inequality problems governed by partial derivatives of the second-order. To this aim, we introduce the approximating solution set and the concept of approximating sequences for the considered controlled variational inequality problem. Further, by using the aforementioned new mathematical tools, we established some theorems on well-posedness. Moreover, the theoretical tools and results included in the paper are accompanied by some examples.