Existence of solutions for a class of singular elliptic systems with convection term Claudianor O. Alves Unidade Acadêmica de Matemática e Estatística Universidade Federal de Campina Grande 58429-900, Campina Grande - PB - Brazil e-mail address:
[email protected]
Abdelkrim Moussaoui
y
Biology Department, A. Mira Bejaia University, Targa Ouzemour 06000 Bejaia, Algeria e-mail address:
[email protected]
Abstract We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines sub and supersolution method with the pseudomonotone operator theory and perturbation arguments involving singular terms. 2000 Mathematics Sub ject Classi…cation: 35J50, 35J60, 35J75 Keywords: pseudomonotone operator, Elliptic Singular Equation, Nonlinear Equations
C.O. Alves was partially supported by CNPq/Brazil 304036/2013-7,
[email protected] y A. Moussaoui was supported by the European program Averroès-Erasmus Mundus(Grant No ]1872)
1
2
1
Introduction
In this work, we focus our attention on the existence of solutions for the following class of elliptic system with convection term 8 u = v 11 v 1 + g1 (ru; rv) in ; > > > > > > > > v = u1 2 u 2 + g2 (ru; rv) in ; < (S) > > u; v > 0 in ; > > > > > > : u = v = 0 on @ ;
where is a bounded domain with smooth boundary and gi : R2N ! [0; +1); i = 1; 2; are positive continuous functions belong to L1 (R2N ). We consider the system (S) in a singular case assuming i ; i ; 2 [0; 1) for i = 1; 2: Hereafter (u; v) is a solution to (S) if u; v 2 C 2 ( ) \ H01 ( ) are both positive in and satisfy the equations of (S) in the classical sense. Nonlinear singular boundary value problems are mathematically challenging and important for applications. They arise in several physical situations such as ‡uid mechanics, pseudoplastics ‡ow, chemical heterogeneous catalysts, non- Newtonian ‡uids, biological pattern formation, for more details about this subject, we cite the papers of Fulks & Maybe [17], Callegari & Nashman [6, 7] and the references therein. Systems (S) can be see as a version of the singular scalar equations 8 u + g(ru) in ; u = u1 > > > > < u > 0 in ; (P ) > > > > : u = 0 on @ ;
with ; > 0 and g : RN ! R be a continuous function verifying some technical conditions. Several works are devoted to classes of problems covering (P ) . For instance, see the papers of Aranda [5], Ghergu & Radulescu [21, 22], Giarrusso & Porru [18], Lair & Wood [25], Zhang [30] and references therein. Problem (P ) without a convection term, that is g = 0 was also investigated. Relevant contributions regarding this situation can be
3 found in Crandall, Rabinowitz & Tartar [9], Choi & McKenna [8], Coclite & Palmieri [10], Círstea, Ghergu & Radulescu [11], Dávila & Montenegro [12] and Diaz, Morel & Oswald [14]. The main tools used in the aforementioned works are Sub and Supersolution, Fixed Point Theorems, Bifurcation Theory and Galerkin Method. On the other hand, using variational technique, more precisely mountains pass theorem, de Figueiredo, Girardi & Matzeu [13] studied a class of elliptic problems without singularity, where the nonlinearity depends of the gradient of the solution. Related to systems (S) , to date, the only case considered in the literature, known to authors for gi 6= 0 is the paper due to Alves, Carrião & Faria [4]. For the case where gi = 0, we refer the reader to the survey paper by Alves & Corrêa [1], Alves, Corrêa & Gonçalves [2], El Manouni, Perera & Shivaji [15], Ghergu [19, 20], Hernández, Mancebo & Vega [23], Montenegro & Suarez [27] and Motreanu & Moussaoui [28]. From the above commentaries, we observe that in recent years singular elliptic problems with convection term has received few attention. Motivated by this fact, our aim is to show the existence of solutions for a class of elliptic systems where the nonlinearity besides a singular term has a convection term. The proof combines results involving pseudomonotone operators, sub and supersolution method and perturbation arguments involving singular terms. We emphasize that our study complete those made in [1], [2] and [28], in the sense that in those papers the authors did not consider the case where the nonlinearity has a convection term, and also [4], because a di¤erent type of singular term was considered. The method used in the present work is di¤erent from those applied in the aforementioned papers. Our main result is the following: Theorem 1.1 Assume that gi : RN ! R are continuous functions and i ; i 2 [0; 1) for i = 1; 2. Then, the systems (S) has a solution. The proof of Theorem 1.1 is done in Sections 3 and 4. The …rst main technical di¢ culty is that the nonlinearities of (S) depend of gradient of the solution, which is more a complicating factor. Indeed, for the scalar case, an interesting result is proved by Kazdan & Kramer [24] and Leon [26], where the authors develop a sub and supersolution method for scalar problem where the nonlinearity depends of the gradient. Instead, the counter-part of this result for systems with gradient terms is not known in the literature. Thus, we do not know a result involving sub and supersolution that could be use to
4 establishes the existence of solution for this class of system. To overcome this di¢ culty, we show in Section 2 a result, see Theorem 2.1, which can be see as a sub and supersolution method for systems whose nonlinearity depends of gradient. The second main di¢ culty in the proof of Theorem 1.1 is associated with the fact that the sub and supersolution method in its version involving maximum principle cannot be used directly for systems involving the gradient of the solution. Moreover, the way as the singularities appear in the system (S) is a di¢ cult point to work with maximum principle. In order to overtake the stated problem we …rst introduce a parameter " > 0 in (S) , giving rise to regularized systems for (S) whose study is relevant for our initial problem. Then, for the regularized systems, we combine variational methods with the sub-supersolution one to prove the existence of a solution (u" ; v" ) 2 H 1 ( )\L1 ( ) H 1 ( )\L1 ( ). This solution (u" ; v" ) is located in some rectangle formed by the sub and supersolution, independent for " > 0, which does not contains zero for all " > 0. Then, a positive solution of (S) is obtained by passing to the limit as " ! 0. This is based on a priori estimates and Hardy-Sobolev inequality. The positivity of the solution is derived from the independence of the subsolution of the regularized systems on ". The rest of this article is organized as follows: In section 2 we state and prove a general theorem about sub and supersolution method for systems with convection term. Sections 3 and 4 contain the proof of Theorem 1.1.
2
An auxiliary result
The main goal in this section is to prove the Theorem 2.1 below, which is a key point in the proof of Theorem 1.1. An interesting point related to Theorem 2.1 is the fact that it is a result of sub and supersolution whose the proof is made using pseudomonotone operator theory. Theorem 2.1 Let H; G : R+ R+ RN RN ! R continuous functions function verifying the following conditions: Given T; S > 0, there exist C > 0 and ; 2 (0; 1), such that jH(x; s; t; ; )j; jG(x; s; t; ; )j
for all (x; s; t; ; ) 2 [0; T ] 1;1 u; u; v; v 2 W ( ) with u(x)
[0; S]
g~(x)
RN
C(1 + j j + j j )
RN . Let g~; gb 2 C 2 ( ) and
u(x) on @
5 and v(x) Assume that
Z
Z
Z
and
Z
ru r dx rv r dx ru r dx rv r dx
gb(x) Z
Z
Z
Z
v(x) on @ :
H(x; u; v; ru; rv) dx; G(x; u; v; ru; rv) dx; H(x; u; v; ru; rv) dx G(x; u; v; ru; rv) dx;
for all nonnegative functions ; 2 H 1 ( ). 1 1 1 1 (u; v) 2 (H ( ) \ L ( )) (H ( ) \ L ( )) verifying u(x)
u(x)
u(x) and v(x) u
and
Z
Z
ru r dx = rv r dx =
Z
Z
g~; v
v(x)
gb 2 H01 ( )
Then,
there is
v(x) 8x 2 ;
H(x; u; v; ru; rv) dx 8 2 H01 ( ); G(x; u; v; ru; rv) dx 8 2 H01 ( );
that is, (u; v) is a solution of the system 8 u = H(x; u; v; ru; rv) in > > > > < v = G(x; u; v; ru; rv) in (AS) > > > > : u(x) = g~; v(x) = gb on @ :
; ;
Proof. Here, we will adapt some arguments found in Leon [26]. Firstly, we introduce two new functions 8 H(x; u(x); v(x); ru(x); rv(x)); s u(x) > > > > < H(x; s; v(x); ; rv(x)); u(x) s u(x) and t v(x) H(x; s; t; ; ); u(x) s u(x) and v(x) t v(x) H1 (x; s; t; ; ) = > > H(x; s; v(x); ; rv(x)); u(x) s u(x) and t v(x) > > : H(x; u(x); v(x); ru(x); ru(x)); s u(x):
6 and 8 G(x; u(x); v(x); ru(x); rv(x)); t v(x) > > > > < G(x; u(x); t; ru(x); ); v(x) t v(x) and s u(x) G(x; s; t; ; ); v(x) t v(x) and u(x) s u(x) G1 (x; s; t; ; ) = > > u(x); t; ru(x); ); v(x) t v(x) and s u(x) G(x; > > : G(x; u(x); v(x); ru(x); rv(x)); t v(x):
Moreover, for each l 2 (0; 1), we consider 1 (x; s)
=
((u(x)
s)+ )l + ((s
u(x))+ )l
2 (x; t)
=
((v(x)
t)+ )l + ((t
v(x))+ )l :
and Using the above functions, we will work with the ensuing auxiliary system
(S1 )
8 > > > > < > > > > :
u = H1 (x; u; v; ru; rv)
1 (x; u)
in
;
v = G1 (x; u; v; ru; rv)
2 (x; v)
in
;
u(x) = g~; v(x) = gb on @ :
Setting the functions
H2 (x; s; t; ; ) = H1 (x; s; t; ; )
1 (x; s)
G2 (x; s; t; ; ) = G1 (x; s; t; ; )
2 (x; t);
and we de…ne the operator B : E ! E 0 given by Z hB(u; v); ( ; )i = (rur + rvr )dx Z
where E = H01 ( )
Z
H2 (x; u; v; ru; rv) dx
G2 (x; u; v; ru; rv) dx;
H01 ( ) is endowed of the norm 1
k(u; v)k = (kuk2 + kvk2 ) 2 ;
7 with k k being the usual norm in H01 ( ). Using the hypotheses on H and G together with the de…nition of H1 ; H2 ; G1 ; G2 ; 1 and 2 , we can prove the ensuing properties for operator B: I) B is continuous: The proof of this property follows by using the fact that H1 ; G1 belong to L1 . II) B is bounded: Here, the boundedness of B is understood in the sense that if U E is a 0 bounded set, then B(U ) E is also bounded. This property also follows using the boundedness of H1 and G1 . III) B is coercive: Here, it is enough to prove that hB(u; v); (u; v)i ! +1 as k(u; v)k ! +1: k(u; v)k
Using again the boundedness of H1 and G1 , we derive hB(u; v); (u; v)i
k(u; v)k2
C1 k(u; v)k
C2 k(u; v)kl+1 :
Thus,
showing that
hB(u; v); (u; v)i k(u; v)k
k(u; v)k
C1
C2 k(u; v)kl ;
hB(u; v); (u; v)i ! +1 as k(u; v)k ! +1: k(u; v)k IV) B is pseudomonotone: First of all, we recall that B is a pseudomonotone operator if (un ; vn ) * (u; v) in E and veri…es lim sup hB(un ; vn ); (un ; vn ) n!+1
(u; v)i
0;
(2.1)
8 then lim inf hB(un ; vn ); (un ; vn ) n!+1
( ; )i
hB(u; v); (u; v)
( ; )i 8( ; ) 2 E:
(2.2)
In our case, the weak limit (un ; vn ) * (u; v) in E yields Z H1 (x; un ; vn ; run ; rvn )(un u) ! 0 and
Z
G1 (x; un ; vn ; run ; rvn )(vn
v) ! 0:
Thereby, the above limits combined with (2.1) load to Z lim sup (run r(un u) + rvn r(vn v))
0;
n!+1
from where it follows that (un ; vn ) ! (u; v) in E: The properties I) IV ) allow us to use [16, Theorem 3.3.6] to conclude that B is surjective. Therefore, there exists (u; v) 2 E such that hB(u; v); ( ; )i = 0 8( ; ) 2 E; implying that (u; v) is a solution of (S1 ). Now, our goal is showing that (i) u
u
u and (ii) v
v
(2.3)
v:
We will show only (i), because the same arguments can be used to prove (ii). Choosing ( ; ) = ((u u)+ ; 0) as a test function, we have Z Z rur(u u)+ = H2 (x; u; v; ru; rv)(u u)+ dx: From de…nition of H2 , Z Z Z rur(u u)+ = H1 (x; u; v; ru; rv)(u u)+ dx
1 (x; u)(u
u)+ dx
9 and so, Z rur(u
u)+ dx =
Z
H(x; u; v; ru; rv)(u
Z
u)+ dx
Since (u; v) is a supersolution, it follows that Z Z rur(u u)+ dx rur(u u)+ dx
Z
(u
u)l+1 + dx:
u)l+1 + dx;
(u
or equivalently, Z
jr(u
2
u)+ j dx
Z
(u
u)l+1 + dx
0;
showing that (u u)+ = 0, from where it follows that u u. To prove that u u, we choose ( ; ) = ((u u)+ ; 0) as a test function. Repeating the above arguments, we get Z Z Z H(x; u; v; ru; rv)(u u)+ dx + (u u)l+1 rur(u u)+ dx = + dx: Since (u; v) is a subsolution, it follows that Z Z Z rur(u u)+ dx rur(u u)+ dx + (u
u)l+1 + dx;
or equivalently, Z
jr(u
2
u)+ j dx
showing that (u u)+ = 0, and so, u of H2 and G2 , it follows that
Z
(u
u)l+1 + dx
0;
u. Combining (2.3) with the de…nition
H2 (x; u; v; ru; rv) = H(x; u; v; ru; rv) and G2 (x; u; v; ru; rv) = G(x; u; v; ru; rv); showing that (u; v) is a solution for system (AS).
10
3
Existence of solution for system (S)
In this section, we will study the existence of solution for the following singular elliptic system 8 u = v 11 v 1 + g1 (ru; rv) in ; > > > > > > > > v = u1 2 u 2 + g2 (ru; rv) in ; < (S) > > u; v > 0 in ; > > > > > > : u = v = 0 on @ ; where is a bounded domain in RN with smooth boundary, gi : R2N ! [0; +1) are positive continuous functions belong to L1 (R2N ) and i ; i 2 [0; 1): Our approach consists in considering for > 0 the approximated system
(S )
8 > > > > > < > > > > > :
u= v=
1 (jvj2 + ) 1 (juj2 + )
v
1
+ g1 (ru; rv) in
;
u
2
+ g2 (ru; rv) in
;
1 2
2 2
u = v = 0 on @ :
For this class of system it is possible to …nd sub and supersolution which do not depend of . For example, by using the positive eigenfunction associated with the …rst eigenvalue, we can …nd easily a subsolution (u; u). To get the supersolution, we observe that any large constant M > 0 can be used to get a supersolution (u; v) = (M; M ) with M > kuk1 . Now, let us consider the functions 1
H (x; s; t; ; ) =
(jtj2
and G (x; s; t; ; ) = which are well de…ned in
+ )
1 2
1 (jsj2
R+
+ )
R+
2 2
RN
t
1
+ g1 ( ; )
s
2
+ g2 ( ; );
RN .
11 By using Theorem 2.1, there exists a positive (u ; v ) verifying system 1. From now on, we denote by (S ) . Set = n1 with any integer n (un ; vn ) the solution (u 1 ; v 1 ). Hence, n
(Sn )
8 > > > > > > < > > > > > > :
n
1
un =
1 (jvn j2 + n )
1 2
vn 1 + g1 (run ; rvn ) in
;
2 2
un2 + g2 (run ; rvn ) in
;
1
vn =
1 (jun j2 + n )
un = vn = 0 on @ :
Once that i ; i 2 [0; 1), i = 1; 2, and gi belongs to L1 (R2N ), it follows that (un ; vn ) is bounded in H01 ( ) H01 ( ). Indeed, since u > 0 and vn
un we have kun k
Z
2
and so, kun k
2
Similarly we derive 2
kvn k
un dx + vn 1
Z
v > 0 in
;
g1 (run ; rvn )un dx
Z
un dx + kg1 k1 v 1
Z
jun j dx:
(3.4)
Z
vn dx + kg2 k1 u 2
Z
jvn j dx:
(3.5)
On the other hand, for i 2 [0; 1), i = 1; 2, we may invoke the Hardy-Sobolev inequality in the form stated in [1, Lemma 2.3] to infer that Z Z vn un dx Cku dx Ckvn k: (3.6) n k and v 1 u 2 Combining (3.4)-(3.6) with Sobolev embedding, it follows that (un ; vn ) is bounded in H01 ( ) H01 ( ). Consequently, we can assume that there is (u; v) 2 H01 ( ) H01 ( ) and Gi 2 L2 ( ) verifying un * u; vn * v in H01 ( );
(3.7)
un ! u; vn ! v in Lp ( ) for all p 2 [1; +1);
(3.8)
12 un (x) ! u(x); vn (x) ! v(x) a.e. in
(3.9)
and gi (run ; rvn ) * Gi in L2 ( ):
(3.10)
Once that u
M and v
un
M a.e in
vn
; for all n 2 N;
the limit (3.9) gives u
M and v
u
M a.e in
v
Recall that (Sn ) entails Z 8 Z 1 > > run r' dx = 1 < 1 2 Z Z (jvn j + n ) 2 1 > > run r dx = : 2 (jun j2 + n1 ) 2
:
vn 1 + g1 (run ; rvn ) ' dx un2 + g2 (run ; rvn )
(3.11)
dx
for all ('; ) 2 H01 ( ) H01 ( ): Setting ('; ) = (un u; vn v) in (3.11) yields Z 8 Z 1 > > run r(un u) dx = vn 1 + g1 (run ; rvn ) (un u) dx < 1 21 2 (jv j + ) n Z Z n 1 > > rvn r(vn v) dx = un2 + g2 (run ; rvn ) (vn v) dx: : 2 (jun j2 + n1 ) 2 We point out that (3.7)-(3.10) implies that Z 1 lim vn 1 + g1 (run ; rvn )(un 1 21 2 n!1 (jvn j + n )
u) dx = 0
and lim
n!1
Z
1 (jun j2 + n1 )
2 2
un2 + g2 (run ; rvn ) (vn
v) dx = 0:
Consequently, lim
n!+1
Z
run r(un
u) dx = lim
n!+1
Z
rvn r(vn
v) dx = 0
13 leading to (un ; vn ) ! (u; v) in H01 ( )
H01 ( ):
(3.12)
This way, passing to relabeled subsequences, we have the limits run (x) ! ru(x) and rvn (x) ! rv(x) a.e in
;
which imply that gi (run ; rvn ) * gi (ru; rv) in L2 ( ) (i = 1; 2):
(3.13)
Now, from (3.12) and (3.13), we may pass to the limit in (3.11) to conclude that (u; v) is a solution for (S) . This completes the proof.
4
Existence of solution for system (S)+
In this section, we will study the existence of solution for the following singular elliptic system 8 u = v 11 + v 1 + g1 (ru; rv) in ; > > > > > > > > v = u1 2 + u 2 + g2 (ru; rv) in < (S)+ > > u; v > 0 in ; > > > > > > : u = v = 0 on @ ; by supposing the same hypotheses of Section 3. The existence of solution for (S)+ can be obtained by using the same arguments explored in the previous section. The unique di¤erence is in the construction of the supersolution that we will work. Here, the idea is the following: Fix R > 0 large enough such that BR (0) and denote by e the unique solution of the problem 8 e = 1; in BR (0) < :
e = 0; on @BR (0):
14 Recalling that gi 2 L1 , for M > kuk1 large enough, a simple computation shows that 8 1 (M e) = M + (M e) 1 + g1 (rw1 ; rw2 ) in ; > (M e) 1 > > > < 1 + (M e) 2 + g2 (rw1 ; rw2 ) in ; (M e) = M (S)+ (M e) 2 > > > > : (M e) > 0 in ; for any w1 ; w2 2 H01 ( ). Thereby, the pairs (u; u) and (u; v) = (M e; M e) satisfy the hypotheses of Theorem 1.1.
Acknowledgement. This work was accomplished while the second author was visiting Laboratoire de Mathématiques, Physique et Systèmes (LAMPS) of Perpignan University, with Averroès fellowship. He wishes to thank Perpignan University for the kind hospitality.
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