Weak solutions for some reaction-diffusion systems with balance law and critical growth with respect to the gradient

This paper is concerned with the existence of weak solutions for 2 x 2 reaction-diffusion systems for which two main properties hold: the positivity of the solutions and the triangular structure. Moreover, the nonlinear terms have critical growth with respect to the gradient.


Introduction
This paper deals with existence results for the following Reaction-Difiusion sys- tem : -0394u = f ( x , u , v , ũ, ṽ) + F ( x ) on 03A9 -0394v = g(x, u, v, ~u, ~v) + G(x) on 8Q (1) u = v = 0 anaO where H is an open bounded subset of ]iN with smooth boundary denotes the Laplacian operator on {1 with Dirichlet boundary conditions.Since we are essentially concerned with systems frequently encountered in applica- tions, we restrict ourself to the case of positive solutions satisfying the trian- gular structure.These two main properties are ensured (respectively) by the following hypotheses f (x,0,v,p,q), g(x,u,0,p,q) ~ 0, FM, GM > 0, for all (u, v, p, q) E R+ x R+ RN RN and f(x,u,v,p,q) + g(x,u,v,p,q) ~ 0 f ( x , u , v,p, q) ~ 0 (3) for all (u, v, p, q) E R+ x R+ RN RN and a.e.x ~ 03A9.
When f and g does not depend on the gradient, an extensive literature has dealt with this kind of problems (especially for the parabolic version), existence results have been given in [9], [10], [13], [14].Excellant surveys treating reaction- diffusion systems include Rothe [16] and Smoller [17].If f and g does depend on the gradient, an existence theorem has been proved in [12], by means of Ll method introduced in [13], when the growth of the nonlinearities with respect to the gradient is subquadratic, namely Our objective is to investigate the case a = 2.This critical growth with respect to the gradient creates some difficulties in the passage to the limit for the ap- proximating problem and the L1 method can not be applied in this case.We adopt a different approach based on techniques introduced in [7] for the case of elliptic equations to deal with exponential test function of the truncations.We refer the reader to [4], [6], [7], [15] for a general survey of this method.Let us point out here that the parabolic version of such systems with L2-data has been recently treated by the same authors, see [2].Typical model where the results of this paper can be applied is the following f on n -won 03A9 ~ u = t; = 0 on ao, where the functions p;, = 1, 2, are nonnegative continuous bounded with re- spect to v such that p~ pi and where 1.1 denotes the euclidian norm in IRN.
We have organised this paper in the following manner.In section 2, we give the precise setting of the problem and state our main result.In section 3, we truncate the system and establish suitable a priori estimates.Finally, we prove the convergence of the truncated problem to some solution of our system.The difficulties in this section are similar to those in [7], [15] and the techniques are of the same spirit.But specific new difficulties owed to the nature of the system must be handled. 2Statement of the result

Assumptions
We first introduce the notion of solution to the problem (1) used here.
Throughout this note we will assume that: H1/ f, g : 03A9 x IR2 x R2N e R are measurable H2/ I, 9 : R2 x R are continuous for almost every x in n.
n ii/ The sequence (un,vn) is relatively compact in W1,q0(03A9) x for all ° .
Proof.i/ Consider the equations satisfied by un and un, we can write u n , v n ~ W 1 , q 0 ( 0 3 A 9 ) ( -gn = dun +G in D'(~).
The fact that f n 0 and F E allows us to conclude |fn| = fn ~F~L1(03A9).
n n Similarly, we get iif This assertion follows by a direct application of a result in [8].ii/ There exists a constant R2 depending on k, and such that iii/ There exists a constant R3 depending on k, iv/ Moreover lim -/ = 0 uni f ormly on n.
Remark 2 The same results can be found in [6] and [15].
Proof.i j a/ We first define the following test-function for every t, h > 0 using the fact that fn 0 and Pt,h(Un) > 0 yield Since -/ 0, we get Thanks to Lebesgue's theorem, we have by passing to the limit as t tends to 0 |fn| ~ |F| .We conclude that lim / = 0 uniformly on n. hoo b/ The main idea is to consider the equation satisfied by 2un + vn, and to take Pt,h(2un + vn) as a test function.We obtain t jv(2~+~-~ Since fn 0, fn +gn 0 and Pt,h (2un + ~n) > 0, we obtain |fn|Pt,h(2un + vn) ~ (2F + G)Pt,h(2un + vn) and / |gn|Pt,h(2un + vn) / (2F + G ) P t , h ( 2 u n + vn) The rest of the proof runs as in the previous step.
ii/ We multiply the first equation in (5) by Tk(un) and we integrate on H, we In the same way, we multiply the second equation in (5) by Tk(vn) and we integrate on S~, we obtain ~|~Tk(vn)|2 = ~gnTk(vn) + ~GTk(vn We then have k(Rl + ~G~L1).
n iii l follows trivially from ii/. iv/We first remark that un satisfies -F, in D'(03A9) if we multiply this inequality by Th(un) and integrate on 03A9, we obtain for every Since un is bounded in L1 (03A9), we have |[un > k]| Ck"1.Therefore, there exists kE independent of n such that ~F~[un>k~] ~ ~ 2 .
n Taking M = kE an letting h tend to infinity, we obtain the desired conclusion.1 The last assertion in lemma 3 allows us to ensure the existence of a subsequence still denoted by (un, un) such that un ~ u in W1,q0(03A9) strongly .
un -u u a.e in Sl.
In the next step, we will show that this subsequence {un, vn) satisfies some useful properties.
Lemma 5 Suppose that vn, u and v are as above.
Then lim I1.3 = 0. Now we investigate 12. Since un and un + vn satisfy the hypotheses of the previous lemma, we get n Then lim ~ = 0 uniformly on n. h~F or the term J1.1, we use Lebesgue's theorem to conclude that lim J1.1 =0.n~F or Ji.2, we write J1.2 = [un~k] Since ~)~~~] converges to 0 a.e., and ~) and is bounded in ~(H), it follows from ([11] lemme 1.3 p12) that converges weakly to 0 in Z~(Q).This implies that the second term of this equality goes to zero as n tends to infinity.Concerning the last term, we remark that |03C6(03BEk,n)|H(un + vn h)~[un~k] 0 a.e. on 03A9 and Lebesgue's theorem, we then have To investigate the remaining term J1.3, we apply Holder's inequality as follows: choose Q such that have Now we use lemma 4-(ii) to obtain J1.3~C(k)(h2~F~L1 Passing to the limit as n tends to infinity (for fixed h, k), the strong convergence of to 0 in L2(O) yields: limsup J1.3 = 0. n~F or the last term J2, we have by a direct application of Lebesgue's theorem In view of inequality (8), we have shown that for k, h fixed lim sup( 7i.2 + 12 -Ji.2) 0.  1 h| ~(un + Vn t.p (03BE'k,n) H( vn + vn h) The proof of this assertion follows closely the steps used in the proof of the previous one, it suffies to replace un by un + vn and u by u + v.It is easily seen that K1 can be treated in the same way as I1.
We have As for the term ~1.3, we write lim K3.2 = 0.

Convergence
The aim of this paragraph is to show that (u, v) obtained before is in fact solution of the problem (I) in the sense of definition 1.By the continuity of the functions f n and gn, we deduce un, vn, ~un, ~vn) ~ f(x, u, v, ~u, ~v) a.e in 03A9.un, vn, ~un, ~vn) ~ g(X, u, v, ~u, ~v) a.e in 03A9.These almost pointwise convergences are not sufficient to ensure that (u, v) is a solution of (1).In fact, we have to prove that the previous convergences are in In view of Vitali's theorem, we have to show that f n and gn are equi-integrable in L1(03A9).Lemma 6 The sequences (fn(x, un, vn, ~un, ~vn))n and (gn(x, un, vn, ~un, ~vn))n are equi-integrable in Ll (03A9).Arguing m the same way as before, we obtain the required result.
by i/ the nonlinearities f n and gn are uniformly bounded in we obtain the required result.. Lemma 4 Let (un , vn) be as in the previous lemma.Then i/ a/ lim |fn| = 0 uniformly on n. b/ lim |fn| = 0 uniformly on n.
k ) |~Tk(un) -~Tk(u)|2H(un + vn h) n nBy using the first assertion (~) and Lebesgue's theorem for the second integral, Proof.Let A be a measurable subset of H, we have| f n ( x , u n , v n , ũ n , ṽ n ) | = |fn| + The only difference is the investigation of the termK3 , indeed