Trapping Region for Discontinuous Quasilinear Elliptic Systems of Mixed
Monotone
Type
By: Siegfried Carl and Joseph W. Jerome
We consider discontinuous quasilinear elliptic systems with
nonlinear boundary conditions of mixed Dirichlet-Robin type on the
individual components. The system considered
is assumed to be of mixed monotone type,
associated with competitive or cooperative species. The vector field
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 are continuous in their off-diagonal (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 fixed point theorem
on complete lattices. Two applications of the theory developed in this
paper are provided. The first application deals with the steady-state
transport of two species of opposite charge within a physical channel, and
in the second application a fluid medium is considered which may undergo
a
change of phase, and which acts as a carrier for certain solute species.
This paper will appear in J. Nonlinear Anal. 51 (2002), 839--859, and
can be viewed in the following format: