CC BY
22
DOI: 10.17516/1997-1397-2020-13-2-151-159 УДК 517.9
Singular Quasilinear Elliptic Systems with (super-) Homogeneous Condition
Hana Didi* Brahim Khodja^
Badji-Mokhtar Annaba University Annaba, Algeria
Abdelkrim Moussaoui*
A. Mira Bejaia University Bejaia, Algeria
Received 02.10.2019, received in revised form 09.12.2019, accepted 16.01.2020 Abstract. In this paper we establish existence, nonexistence and regularity of positive solutions for a class of singular quasilinear elliptic systems subject to (super-) homogeneous condition. The approach is based on sub-supersolution methods for systems of quasilinear singular equations combined with perturbation arguments involving singular terms.
Keywords: singular system, p-Laplacian, sub-supersolution, regularity.
Citation: H.Didi, B.Khodja, A.Moussaoui, Singular Quasilinear Elliptic Systems with (super-) Homogeneous Condition, J. Sib. Fed. Univ. Math. Phys., 2020, 13(2), 151-159. DOI: 10.17516/1997-1397-2020-13-2-151-159.
Introduction
We consider the following system of quasilinear and singular elliptic equations:
—AP1 u1 = XuI1 uf2 + Sh1(x) in Q pp) ^ -\2U2 = Xua2uf2 + Sh2(x) in Q mi, U2 > 0 in Q mi, u2 =0 on dQ
where Q is a bounded domain in RN (N ^ 2) having a smooth boundary dQ, X > 0, S ^ 0 are parameters, and hi e L^(Q) is a nonnegative function. Here APi stands for the p-Laplacian differential operator with 1 < pi < N. A solution of (P) is understood in the weak sense, that is, a pair (ui,u2) e W0)'P1 (Q) x Wg'P2(Q), which are positive a.e. in Q and satisfying
/ \Vui\Pi-2VuiVpi dx = (Xu^+ Shi)pi dx, for all pi e Wl'Pi(Q), i = 1,2. JQ JQ
We consider the system (P) in a singular case assuming that
0 <a2 <p\ - 1, 0 < ¡31 <pl - 1 and - 1 < a1, ¡32 < 0, (1)
* [email protected] [email protected]
^[email protected] https://orcid.org/0000-0003-1336-2257 © Siberian Federal University. All rights reserved
Npi
where p* = —-. This assumption makes system (P) be cooperative, that is, for u^resp. u2)
N - pi
fixed the right term in the first (resp. second) equation of (P) is increasing in u2 (resp. ui).
Recently, singular cooperative system (P) with S = 0 was mainly studied in [8,9,20]. In [20] existence and boundedness theorems for (P) was established by using sub-supersolution method for systems combined with perturbation techniques. In [8] one gets existence, uniqueness, and regularity of a positive solution on the basis of an iterative scheme constructed through a sub-supersolution pair. In [9] an existence theorem involving sub-supersolution was obtained through a fixed point argument in a sub-supersolution setting. The semilinear case in (P) (i.e. pi = 2) was considered in [7, 13, 21] where the linearity of the principal part is essentially used. In this context, the singular system (P) can be viewed as the elliptic counter-part of a class of Gierer-Meinhardt systems that models some biochemical processes (see, e.g. [21]). It can be also given an astrophysical meaning since it generalizes to the system the well-known Lane-Emden equation, where all exponents are negative (see [7]). For the one dimentional case (N = 1) we quote [15] and the references therein. The complementary situation for the system (P) with respect to (1) is the so-called competitive system, which has recently attracted much interest. Relevant contributions regarding this topic can be found in [9,18,19]. For the regular case in (P), that is when all the exponents are positive, we refer to [6,22], while for quasilinear systems with singular weights we cite [2,4] and their references.
It is worth pointing out that the aforementioned works have examined the subhomogeneous case © > 0 of singular problem (P) where
© = (pi - 1 - ai)(p2 - 1 -¡2) - ¡W (2)
The constant © is related to system stability (P) that behaves in a drastically different way, depending on the sign of ©. For instance, for © < 0 system (P) is not stable in the sense that possible solutions cannot be obtained by iterative methods (see [5]).
Unlike the subhomogeneous case © > 0 studied in the above references, the novelty of this paper is to establish the existence, regularity and nonexistence of (positive) solutions for singular problem (P) by processing the two cases: 'homogeneous' when © = 0 and 'superhomogeneous' if © < 0. It should be noted that throughout this paper, © < 0 (resp. = 0) means that Pi - 1 - ai - Pi < 0 (resp. = 0).
The existence result for problem (P) is stated as follows.
Theorem 1. Assume (1), © < 0 (resp. © = 0) and suppose that
inf h1 (x), inf h2(x) > 0. (3)
Q Q
Then, there is S0 > 0 (resp. So,Ao > 0) such that, for all S € (0, So), problem (P) possesses a (positive) solution (u1,u2) in Cq'3(Q) x Cq'3(Q), for certain P € (0,1), verifying
ui ^ cd(x) in Q,
for some constant c > 0 and for all A > 0 (resp. A € (0, A0)). Moreover, if © = S = 0 and
p2 pi ¡1 = — (Pi - 1 - ai) or a2 = — (p2 - 1 - ¡2), (4)
pi p2
then, there exists A* > 0 such that problem (P) has no solution for every A € (0, A*).
The main technical difficulty consists in the presence of singular terms in system (P) with (1), expressed through (super-) homogeneous condition. Our approach is chiefly based on sub-supersolution method in its version for systems [3, section 5.5]. However, this method cannot
be directly implemented due to the presence of singular terms in (P) under assumption (1). So, we first disturb system (P) by introducing a parameter e > 0. This gives rise to a regularized system for (P) depending on e whose study is relevant for our initial problem. By applying the sub-supersolution method, we show that the regularized system has a positive solution (u\e, u2,e) in C1,3(Q) x C13(Q) for some P € (0,1). It is worth noting that the choice of suitable functions with an adjustment of adequate constants is crucial in order to construct the sub-supersolution pair as well as to process the both cases © < 0 and © = 0. The (positive) solution (u1,u2) in (Wo1,P1 (Q) n L~(Q)) x (Wo1,P2 (Q) n L~(Q)) of (P) is obtained by passing to the limit as e ^ 0. This is based on a priori estimates, Fatou's Lemma and S+-property of the negative pi-Laplacian. The positivity of the solution (u1,u2) is achieved through assumption (3) while C1,3-regularity is derived from the regularity result in [11].
The rest of the paper is organized as follows. Section 1 is devoted to the existence of solutions for the regularized system. Section 2 established the proof of the main result.
1. The regularized system
Given 1 < p < the space Lp(Q) and W(j (Q) are endowed with the usual norms
/ \ 1/p ( \ 1/p ||u|l = i J \u\p dxj and ||u||1p = i j\Vu\P dx] , respectively. We will also use the space
Q ' Q '
C13(Q) = {u € C13(Q) : u = 0 on dQ} for a suitable P € (0,1).
In what follows, we denote by the positive eigenfunction associated with the principal eigenvalue A1,pi, characterized by the minimum of Rayleigh quotient
||Vuip
uiew1'Pi(Q)\{0>
xi,Pi = inf "fi—ilpf1 • (5)
For a later use recall there exist constants li,li > 0 such that
11&1,pi (x) ^ 01,p2 (x) ^ 12$1,p1 (x) and l1d(x) ^ (x) ^ l2d(x) for all x € Q, (6)
where d(x) := dist(x,dQ) (see, e.g., [10]).
Let Q be a bounded domain in RN with a smooth boundary dQ such that Q C Q. Denote d(x) := d(x, dQ). By the definition of Q there exists a constant p > 0 sufficiently small such that
d(x) > p in Q. (7)
Define wi € C 1(QQ) the unique solution of the torsion problem
-APiwi = 1 in Q, wi =0 on dQ, (8)
satisfying the estimates
wi(x) ^ cod(x) in Q, (9)
for certain constant c0 € (0,1) (see [12, Lemma 2.1]). For a real constant C > 1, set
(uiF,ui) = (ce^1,pi ,C-1wi), i = 1, 2, (10)
where ce > 0 is a constant depending on e > 0 such that
0 < ce <col-1 C-1. (11)
Then, by (10), (6) and (8), it is readily seen that
ui(x) = C-1wi(x) ^ C-1CQd(x) ^ C-1cqd(x) ^ ^ l-1 C-1CQ^i'Pi (x) ^ ce^iPi (x) = u F(x) in Q, for i = 1, 2.
For every e e (0, eQ), with eQ < 1, let introduce the auxiliary problem
-AP1 u1 = X(u1 + e)ai (u2 + e)31 + Sh1(x) in Q (Pe) { -Ap2U2 = X(ui + e)a2 (U2 + e)32 + Sh2(x) in Q , u1,u2 =0 on dQ
which provides approximate solutions for the initial problem (P).
Lemma 1. Assume (1) and h1, h2 =0 in Q. Then, if © < 0 (resp. © = 0), there is a constant SQ > 0 (resp. SQ,XQ > 0 ) such that for all S e (0, SQ), (u1,u2) in (10) is a supersolution of (Pe) for all X > 0 (resp. X e (0, XQ)) and all e e (0, eQ).
Proof. Assume © < 0 and set eQ = C-1,
sq = 2 c Pi-i\hi\\J. (12)
On account of (1), (7)-(10) and (12), for all S e (0, SQ) and e e (0,eQ), one derives
(ui + e)-a1 (U2 + e)-31 (—Ap1 ui - Shi) > u-a1 (M2 + eQ)-31 (-Ap,ui - S \\hi\\TO) >
> Ca1+31 (cQd(x))-a1 + 1)-31 (C-(p1-1) - Sq \\hi\U >
^ C^1-(P1-1-a1)(cQP)-a1 (\\W2\\TO + 1)-31 (1 - SQCP1-i \\hi\\J >
> 2C^1-(P1-1-a1)(cQp)-a1 (\W2\TO + 1)-31 > X in Q, and similarly
(ui + e)-a2 (U2 + e)-32 (-Ap2 U2 - Sh2) > (Mi + eQ)-a2u-32 (^ U2 - S \h2\J >
> Ca2+32 (\\wi\\TO + 1)-a2 (cQd(x))-32 (C-(p2-1) - Sq \\h2\U = = Ca2-(p2-1-32)(\\wi\\+ 1)-a2(cqp)-32 (1 - SqCp2-1 \\h2\U >
> iCa2-(p2-1-l2)(\wi\^ + 1)-a2(cqp)-32 > X in Q,
for all X > 0, provided C > 1 is sufficiently large. This shows that (u1,u2) is a supersolution pair for problem (P£). If © = 0, by repeating the argument above, the same conclusion can be drawn for X e (0, X0) with a constant X0 > 0 that can be precisely estimated. This completes the proof. □
Lemma 2. Assume (1) and © ^ 0 hold. Then, (m1 e,U2 £) is a subsolution of (Pe) for all X,S > 0 and every e e (0, eQ).
Proof. Fix e e (0,eQ). From (10) and (1), we obtain
(Mi 'e + e)-ai (u.2'e + e)-31 (-Ap1 ui'e - Shi) <
< cP1-1(cE^ipp1 + eQ)-a1 (ce^i'P2 + e)-31 Xip fl1-1 <
.. (13)
< cp1-1(^i,p1 + eQ)-a1 (c^i'P2 + e)-31 Xip14>P-p\ <
< cp1-1e-31 (Uip1 ^ + 1)-a1 Xi'P1 Ui,p1 \\P1-1 < X in Q
and similarly
Ke + e)-a'2 (U2,e + e)-ß2 (-Ap2u2 e - 5h2) <
< (uM + e)-a2(U2,e + £o)-ß2(Ap2U2,e) =
= cPe2-1(ce^i,Pl + e)-a2 (^i,P2 + eo)-ß2 Ai,p2 tip2- <
< cp2-1e-a2 (y^i,P2 \L + eo)-ß2 Ai,p2 W^ip\C-1 < A in Q,
(14)
provided cE > 0 is sufficiently small. Gathering (13) and (14) together yields
—APiui E ^ A(ui E + £)ai(u2 E + e)3i + Shi in Q, proving that (u1e,u2e) in (10) is a subsolution pair for problem (PE). □
We state the following result regarding the regularized system. Theorem 2. Assume (1) and h1,h2 =0 in Q. Then
(a) If © < 0 (resp. © = 0) there exist a constant S0 > 0 (resp. S0, Ao > 0) such that for all
S G (0, So) system (PE) has a (positive) solution (u1E, u2,E) G C^'3(Q) x C0'3(Q), ¡3 G (0,1), satisfying
uiE(x) ^ ui(x) in Q, (15)
for all A > 0 (resp. A G (0, A0)), and every e G (0, e0).
(b) For © ^ 0 and under assumption (3), if S > 0, there exists a constant c0 > 0, independent
of e, such that all solutions (u1e,u2e) of system (PE) verify
ui'£(x) ^ c0d(x) for a.a. x G Q, for all e G (0,e0). (16)
Proof. On the basis of Lemmas 1 and 2 together with [3, section 5.5] there exists a solution (u1,e,u2,e) of problem (PE), for every e G (0, e0). Moreover, applying the regularity theory (see [16]), we infer that (u1 u2, E) G C0' 3(Q) x C0' 3(Q) for a suitable 3 G (0,1). This proves (a).
Now, according to (3), let a > 0 be a constant such that infq h1(x), infq h2(x) > a. Define zi the only positive solution of
—APizi = Sa in Q, zi = 0 on dQ,
which is known to satisfy zi(x) ^ c2d(x) in Q. Then it follows that —APiuE ^ — APizi in Q, uiE = zi on dQ, for all e G (0, e0), and therefore, the weak comparison principle ensures the assertion (b) holds true. □
2. Proof of the main result
Set e = — with any positive integer n > 1/e0. From Theorem 2 with e = —, there exists n n
ui'n := u'n such that
—Ap.ui'n, <fi) = A j [u1' n + ^ (u-2n + n) ^i dx + S j hi^idx, (17)
for all pi e W0'Pi(Q), i =1, 2. Taking p 1 = u 1 ,n in (17), since a 1 < 0 < 31, we get \ui,n\piP1 = X/ (uin + 1) \V'2,n + -) Uin dx + I ShiUin dx <
1 • p>lv1 ,^ n v2 • .o
< X ul 1r+l (u2, n + 1)ßl dx + S 11^1 ||œ / ui, n dx <
JO ' Jo (18)
< X [ ü^1 (u2 + 1)ßl dx + S llh^lf u1 dx <
Jo J o
< x|Qi(iü1iai+1 (iML+i)ßi+s IMU MU ).
Hence, {u1'n} is bounded in Wi'P1 (Q). Similarly, we derive that {u2j} is bounded in W0 'P2 (Q). We are thus allowed to extract subsequences (still denoted by {u^}) such that
uin ^ ui in WiPi (Q), i = 1, 2. (19)
The convergence in (19) combined with Rellich embedding Theorem and (15)-(16) entails
CQd(x) ^ ui(x) ^ ui(x) in Q. (20)
Inserting pi = ui,n - ui in (17) yields
1 \a. ( 1 \3i
{ Api ui, ni ui, n ui)
x(u1, n + (u2, n + + Shi
V n/\ n/
(ui,n - ui) dx.
We claim that
lim {-APiui n, ui n - ui) < 0.
<
Indeed, from (15), (16) and (10), we have
^ (uin + (u2,n + n) + Shi^j (ui,n - Ui)
< Un (U2,n + 1)31 + Shi)(\uin\ + \ui\) <
< 2((cqd(x))a1 (U2 + 1)31 + Shi)ui <
< 2((cqd(x))a1 (\\M2\\TO + 1)31 + S\hi\x ) \\Mi\TO < CQd(x)a1 in Q,
with some positive constant CQ. Then, (1) together with Lemma in [14, page 726] imply that
, ri + n) (u2,n + n) + Sh^j (u1,ri - u1) G L1(Q).
(21)
Using (19), (21) and applying Fatou's Lemma, it follows that
a1 / i \ 31
lim sup / ( (ui n + 1) (U2 n + -) + Shi) (ui n - ui) dx <
n—ao Jq V V nJ V nJ J
< J nl—m^sup yip ^Ui ' n + (u'2' n + + Sh^j (ui' n - ui)^ dx ^ 0,
showing that lim sup (-AP1 ui n,ui n - ui) < 0. Likewise, we prove that
lim sup (-Ap2U2'n,U2'n - U2) < 0.
o
Then the S+-property of —APi on W^'Pi (Q) (see, e.g., [17, Proposition 3.5]) guarantees that
ui,n —> ui in w0'Pi(Q), i = 1, 2. On account of (17), besides (22), the next step is to verify that
lim
n^oo
I (ui,n + -1 (u2,n + -1 Vi dx = I uaußiVi dx,
/n v nJ v nJ Jq
(22)
(23)
for all Vi G Wi,pi(Q). By (15), (16), (1) and (20), it holds
/ 1 / 1 \ßi
Ui,n + -) U2,n + - Vi V n/\ nJ
and
Ui,n + -
1 \ "2
U2,n + - I n
1 \ß2
V2
< (cod(x))ai (\\U2\L + 1)ß1 Ivil
< (ML + 1)a2(cod(x))ß21V21•
Then, by (1) together with Hardy-Sobolev inequality (see, e.g., [1, Lemma 2.3]), assertion (23) stem from Lebesgue's dominated convergence Theorem. Hence we may pass to the limit in (17) to conclude that (u1,u2) is a solution of problem (P) satsifying (20). Furthermore, using (1), (20) and (10), one has
ui1 uß1 + Shi < ui1 uß1 + S \\hi\\TO <
< (cod(x)r1 \\v\t + S \\hi\L d(x)a^ <
< Cid(x)a1 for all x G Q
(24)
and
u?2 uß2 + Sh2 < ür2 uß2 + S
<
< U^C (cod(x)f2 + S d(xf-2 < (25)
< C2d(x)^2 for all x G Q,
for certain positive constants C{ and C2. Hence, (1) enable us to apply Lemma 3.1 in [11] to infer that (u,v) G C^(Q) x C0^(Q) for some 3 G (0,1).
We are left with the task of determining the nonexistence result stated in Theorem 1. Arguing by contradiction and assume that (u,u2) is a positive solution of problem (P) with S = 0. Multiplying in (P) by ui, integrating over Q, applying Young inequality with ai,/2 > -1, we get
[ IVuiIP1 dx = A f u^+uß1 dx < A f (
Jn Jn Jn V
ai + 1 Pi
uP1 +
pi — 1 — ai
-1
Pi
dx
(26)
and
IVu2lP2 dx = A I uf uß2+1dx < All P--^^ uf-2-?2 + uP2 ) dx. (27)
p2
p2
Adding (26) with (27), according to (4), this is equivalent to
\\Vui\\P1 + \\Vu2\\P2 < A
p2
ai + 1 P2 — 1 — ß2
Pi
■ +
p2
\ui\P1 +
ß2 + 1 , Pi — 1 — ai
p2
+
Pi
\M\
P2
. (28)
Since © = 0, observe from (4) that
ai + 1 + P2 — 1 — ß2 = ai + a2 + 1
Pi
p2
ß2 + 1 + Pi — 1 — ai
p2
Pi
Pi
ßi + ß2 + 1 P2 '
1
2
n
n
n
Then gathering (5), (28) and (29) together yields
ai + a2 + 1,\,, p , ßi + ß2 + 1
X1,pi -
P1
which is a contradiction for
X llu1llpp1 + (\ltp2 - ß1 +P22 + 1 x) ||u2|P2 < 0
\ ^ \ ■ Pi \ P2 \
\<\* = mm < ---—-Xi,Pl, Xi,P2
{ ai + a2 + 1 ßi + ß2 + 1
Thus, problem (P) has no solution for A < A*, which completes the proof.
References
[1] C.O.Alves, F.J.S.A.Correa, On the existence of positive solution for a class of singular systems involving quasilinear operators, Appl. Math. Comput., 185(2007), 727-736.
[2] S.Boulaaras, R.Guefaifia, T.Bouali, Existence of positive solutions for a class of quasilinear singular ellipticsystems involving Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions, Indian J. Pure Appl. Math., 49(2018), no. 4, 705-715.
[3] S.Carl, V.K.Le, D.Motreanu, Nonsmooth variational problems and their inequalities. Comparaison principles and applications, Springer, New York, 2007.
[4] Z.Deng, R.Zhang, Y.Huang, Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity, Math. Meth. App. Sci., 40(2017), no. 5, 1538-1552.
[5] P.Clement, J.Fleckinger, E.Mitidieri, F.de Thelin, Existence of Positive Solutions for a Non-variational Quasilinear Elliptic System, J. Diff. Eqts., 166(2000), 455-477.
[6] C.Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonl. Anal., 62(2005), no. 4, 751-756.
[7] M.Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258(2010), 3295-3318.
[8] J.Giacomoni, J.Hernandez, A.Moussaoui, Quasilinear and singular systems: the cooperative case, Contemporary Math., 540(2011), 79-94.
[9] J.Giacomoni, J.Hernandez, P.Sauvy, Quasilinear and singular elliptic systems, Advances Nonl. Anal., 2(2013), 1-41.
[10] J.Giacomoni, I.Schindler, P.Takac, Sobolev versus H older local minimizers and existence of multiple solutions for a singular quasilinear equation, A. Sc. N. Sup. Pisa, 6(2007), no. 5, 117-158.
[11] D.D.Hai, On a class of singular p-Laplacian boundary value problems, J. Math. Anal. Appl., 383(2011), 619-626.
[12] D.D.Hai, H.Wang, Nontrivial solutions for p-Laplacian systems, J. Math. Anal. Appl., 330(2007), 186-194.
[13] J.Hernandez, F.J.Mancebo, J.M.Vega, Positive solutions for singular semilinear elliptic systems, Adv. Diff. Eqts., 13(2008), 857-880
14] A.C.Lazer, P.J.Mckenna, On a singular nonlinear elliptic boundary-value problem, Proc. American Math. Soc., 111(1991), no. 3, 721-730.
15] Y-H.Lee, X.Xu, Global Existence Structure of Parameters for Positive Solutions of a Singular (pi,p2)-Laplacian System, Bull. Malays. Math. Sci. Soc., 42(2019), no. 3, 1143-1159.
16] G.M.Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12(1988), 1203-1219.
17] D.Motreanu, V.V.Motreanu, N.S.Papageorgiou, Multiple constant sign and nodal solutions for Nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10(2011), no. 5, 729-755.
18] D.Motreanu, A.Moussaoui, A quasilinear singular elliptic system without cooperative structure, Act. Math. Sci., 34(2014), no. 3, 905-916.
19] D.Motreanu, A.Moussaoui, An existence result for a class of quasilinear singular competitive elliptic systems, Applied Math. Letters, 38(2014), 33-37.
20] D.Motreanu, A.Moussaoui, Complex Variables and Elliptic Equations, 59(2014), 285-296. DOI: 10.1080/17476933.2012.744404
21] A.Moussaoui, B.Khodja, S.Tas, A singular Gierer-Meinhardt system of elliptic equations in RN, Nonlinear Anal., 71(2009), 708-716.
22] R.S.Rodrigues, Positive solutions for classes of positone/semipositone systems with multiparameters, Electronic Journal of Differential Equations, (2013), no. 192, 1-10.
Сингулярные квазилинейные эллиптические системы с (супер-)однородным условием
Хана Диди Брахим Ход
Университет Баджи-Мохтар Аннаба Аннаба, Алжир
Абделькрим Муссауи
Университет Мира Беджайа Беджайа, Алжир
Аннотация. В данной работе мы устанавливаем существование (несуществование) и регулярность положительных решений для класса сингулярных квазилинейных эллиптических систем, подчиняющихся (супер-)однородному условию. Подход основан на методах субсуперрешений для систем квазилинейных сингулярных уравнений в сочетании с аргументами возмущения, включающими сингулярные члены.
Ключевые слова: сингулярная система, р-лапласиан, субсуперрешение, регулярность.
