On $T$-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data

We introduce the notions of $T$-solutions and shift $T$-solutions of variational inequalities corresponding to a non-linear degenerate anisotropic elliptic operator, a constraint set in a sufficiently large class, and an $L^1$-right-hand side. We prove theorems on the existence and uniqueness of such solutions and describe their properties. While the notion of $T$-solution is defined only when the constraint set contains at least one bounded function, the notion of shift $T$-solution does not require this condition. We describe the relation between these notions and prove that these types of solutions of a variational inequality coincide with ordinary solutions whenever the right-hand side is sufficiently regular.