Subsolution-supersolution method in variational inequalities (Q5945993)

The subsolution-supersolution method for equations is extended to a class of elliptic variational inequalities of the type NEWLINE\[NEWLINE\int_\Omega A(x,\nabla u) \cdot(\nabla v-\nabla u)\;dx\geq \int_\Omega F(x,u)(v-u)\;dxNEWLINE\]NEWLINE \(\forall v\in K, \;K\subset W^{1,p}(\Omega),\) closed convex. Under additional assumptions on \(K\), the author proves the existence of a maximal and a minimal solution. The proofs rely mainly on the lattice structure of the Sobolev space \(W^{1,p}(\Omega)\). The obtained results are also of interest in view of the fact that the solution set of variational inequalities is in general more complicated than the solution set of equations.