COMPACTNESS OF SUPPORT OF SOLUTIONS FOR SOME CLASSES OF NONLINEAR ELLIPTIC AND PARABOLIC

In this paper, we obtain some existence Theorems of nonnegative solutions with compact support for homogeneous Dirichlet elliptic problems; we extend also these results to parabolic systems.

Supersolution and comparison principles are our main ingredients. 


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. Introductio n
This paper is concerned with the existence of nonnegative solutions with compact support in X := Wó ' p (SZ ) n ( SZ ) x Wó' q ( SZ ) n (12) fo r the following systems : _Opu+alu~a-1u = f(x,u,v) in 9 (S) -L] q v + blvi a -1 v = g(x, u, v) in 9 u=v =o on 09 , and next 9 at ---q pu + alui'u = f (x, u, v) in SZ x R + át -A q v + blv where p ~1, q ~1, a, b, rx and fi are positive constants ; the operator qp u, defined by qp u := div(1Vui p-2 Vu), p ~1 is the well known "p-Laplacian" ; f f and g are nonnegative Caratheodory functions and uo an d vo are some given functions .
During recent years, many papers are devoted to the study of reaction- diffusion systems which arise very often in applications such as, mathematical biology, chemical reactions and combustion theory .An excellent overview of the subject is the survey of [1] .
Díaz and Herrero [4] and [51, study the case of a single equation of the form : _pp u --}-alul°`T1 u = f in SZ (Ea,,g) u = g on r7 g , where a is a positive constant, f E L'(Q), g E W l,p (9) and glan E L'(8Q), both with compact support .Then, a necessary and sufficien t condition for the existence of a solution u E W 1 ' p (1-2) n L°° (S?) of (Ea ,j ) with compact support is o < a C p -1 .They obtain the same results for the associated parabolic problem .
Here we generalize the aboye resuits to some elliptic and paraboli c systems and we used the iterative method based on the Compariso n Principie for the problem (Ti ), taking in account the construction of subsuper solution introduced in [111 .
Our paper is organized as follows :

. Preliminaries
We shall use the following notations : For p 01, +oo[, p * is defined by p + p* = 1 .For a> o,p> Consider the following sets : where B(R) :_ {x E /IxI < R} and A' is the complement any set o f the A .
I} A pair (ü, '13) -í;) is said to be a weak sub-super solution for th e Dirichlet problem (S) if the following conditions are satisfied : Similar definitions can be found in Díaz-Hernández [3], Hernández [7] .
In this paper we also use the following lemmas : Lemma 3 .[8] Let Y be a Banach space .
If u E L P (O,T ;Y) and á E LP (O,T ;Y)(1 < p < +oo) .Then alter an eventual modification on a set of measure zero of (0, T), u is continuous from [0, T] to Y .
First we need that ic E e1 ~SZ} (resp .v E C1 ( SZ) } which implies that th e . These constants will be completely determined in each one of the following sections .
(7-12 ) The functions f (x, u, v) and g(x, u, v) satisfy : We seek solutions (u, v) E X satisfying (S) in the distributional sense.
(3 .16) A similar argument can be used for z .Since K is a convex, bounded and closed subset of E, we can apply Schauder ' s Fixed Point Theorem and obtain the existence of a fixed point for T, which gives the existence o f at least one solution (u, v) of (S) such that o Ç u ç ic and o C v f) .■
Proof: Using an iterative method, we proceed in five steps .
We extend u by 0 and v by 0 out of S2 x [0, T] (VT > 0) .
v) Uniqueness : Suppose that there exists (u l , U2) and (v i , v 2 ) two solutions of problem (P), we have : multiplying by (u i -u2 ) , we obtain by using the monotonicity of operato r -Op and (N3 ) : 1 a -( u 1 -U2) 2 (X, t) dx dt Ç K B,B2 where C is a positive constant .

and 1 ( 2 )
Q, o) -is a sub-super solution of (P) .Proof: From the definition of ic, for (II-b) it is sufficient to have : IIuoIIo c xl xi-c -1 .