ABSTRACT
Chapter 5
Extremality results for
quasilinear PDE
The upper and lower solution method combined with the monotone iteration
technique has been proved to be a powerful tool to obtain the existence
of extremal solutions of elliptic and parabolic boundary value problems
within an order interval [u; u] formed by an ordered pair of upper and lower
solutions u and u, respectively. However, in order to apply the monotone
iteration technique to such type of problems given in an abstract form by
Au = Fu; the operator A which stands for an elliptic or parabolic operator
related with some boundary and initial conditions has to have an increasing
inverse, and the operator F which stands for the Nemytskij operator of
the lower order terms has to be increasing with respect to the underlying
natural partial ordering of functions. For example, semilinear elliptic and
parabolic problems with a nonlinear lower order term depending only on
u which is either an increasing or a Lipschitz continuous function of u can
always be transformed by the maximum principle into an abstract setting
given above. Thus the above problem may be rewritten as a xed point
equation
u = A
Fu
within some function space governed by an increasing xed point operator
A
F : [u; u]! [u; u] of the order interval [u; u] to itself which allows one
to obtain the extremal solutions within [u; u] by successive approximation.
