Monotone Penalty Approximation of Extremal Solutions for Quasilinear Noncoercive Variational Inequalities

This paper is about a monotone approximation scheme for extremal (least or greatest) solutions of the following variational inequality: u set membership, variant K: left angle bracket Au+F(u), v−uright-pointing angle bracket > or =, slanted 0, for all v set membership, variant K, in the interval between some appropriately defined sub- and supersolutions. The variational inequality is approximated by a sequence of penalty equations. The extremal solutions of the penalty equations, constructed iteratively and forming a monotone sequence, are proved to converge to the corresponding solutions of the original inequality. We note that no monotoneity assumption on the lower-order term F is imposed.