Trapping region for discontinuous quasilinear elliptic systems of mixed monotone type

Abstract

We consider discontinuous quasilinear elliptic systems with nonlinear boundary conditions of mixed Dirichlet-Robin type on the individual components. The system considered is of the general form Au + f(¢;u )= h, where A is a quasilinear elliptic operator of Leray-Lions type, u =( u 1;u2);and the vector fleld f =( f 1;f2 )i s assumed to be of mixed monotone type associated with competitive or cooperative species. The vector fleld f may be discontinuous with respect to all its arguments. The main goal is to prove the existence of solutions within the so-called trapping region. Furthermore, if, in addition, the components fk are continuous in their ofidiagonal (nonprincipal) arguments, one can show the compactness of the solution set within the trapping region. The main tools used in the proof of our main result are variational inequalities, truncation and comparison techniques employing special test functions, and Tarski’s flxed point theorem on complete lattices. Two applications of the theory developed in this paper are provided. The flrst application deals with the steady-state transport of two species of opposite charge within a physical channel, and in the second application a ∞uid medium is considered which may undergo a change of phase, and which acts as a carrier for certain solute species.