- Email: [email protected]
Existence and non-existence of solutions for an elliptic system
Applied Mathematics Letters 37 (2014) 118–123
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Existence and non-existence of solutions for an elliptic system Dragos-Patru Covei ∗ Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st District, Postal Code: 010374, Postal Office: 22, Romania
article
info
abstract
Article history: Received 24 March 2014 Received in revised form 9 June 2014 Accepted 10 June 2014 Available online 21 June 2014
We study the existence of positive solutions for a system of two elliptic equations of the form
−∆u = a1 (x)F1 (x, u, v) −∆v = a2 (x)F2 (x, u, v) u = v = 0 on ∂ Ω
Keywords: Elliptic systems Existence Non-existence
in Ω in Ω
where Ω ⊂ RN (N ≥ 2) is a bounded domain in RN with a smooth boundary ∂ Ω or Ω = RN (N ≥ 3). A non-existence result is obtained for radially symmetric solutions. Our proofs are based primarily on the sub and super-solution method. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the existence and non-existence of positive solutions for the possibly singular Dirichlet problem for the Lane–Emden–Fowler system
−∆u = a1 (x) F1 (x, u, v) in Ω , −∆v = a2 (x) F2 (x, u, v) in Ω , u = v = 0 in ∂ Ω ,
(1.1)
where Ω ⊂ RN (N ≥ 2) is a bounded domain with a smooth boundary ∂ Ω or Ω = RN (N ≥ 3). (If Ω = RN , then the condition u = v = 0 on ∂ Ω should be understood as u (x), v (x) → 0 as |x| → ∞). Each ai is a positive C 0,α (Ω ) (α ∈ (0, 1)) function, and the function Fi : Ω × (0, ∞) × (0, ∞) → (0, ∞) is continuously on its domain and satisfies the following: (F1) Fi is locally Hölder for each i; g (s) (F2) for every i ∈ {1, 2} there exists a continuous function gi : (0, ∞) → (0, ∞), for which i s is decreasing on (0, ∞) and lims→∞ gi (s) /s := 0, such that Fi (x, t1 , t2 ) ≤ gi (ti ) for all (x, t1 , t2 ) ∈ Ω × (0, ∞) × (0, ∞); (F3) for every i ∈ {1, 2} there exists hi : (0, ϑ1 ) × (0, ϑ2 ) → (0, ∞) continuous non-increasing function such that Fi (x, t1 , t2 ) ≥ hi (t1 , t2 ) for all (x, t1 , t2 ) ∈ Ω × (0, ϑ1 ) × (0, ϑ2 ), for some ϑi ∈ (0, 1) and lims→0 hi (s, s) ∈ (0, ∞]. The scalar case corresponding to the system (1.1)
−∆u = F (x, u) in Ω , ∗
u=0
on ∂ Ω
Tel.: +40 0766224814. E-mail addresses: [email protected], [email protected].
http://dx.doi.org/10.1016/j.aml.2014.06.007 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
(1.2)
D.-P. Covei / Applied Mathematics Letters 37 (2014) 118–123
119
has been studied by several authors and applied to a variety of problems. Examples are boundary layer phenomena for viscous fluids (Callegari–Nashman [1], Shaker [2]), pseudoplastic fluids (Callegari–Nashman [3]), reaction–diffusion processes and chemical heterogeneous catalysts (Aris [4]) and heat conduction in electrically conducting materials (Diaz [5]). Others have studied similar problems with less focus on specific applications such as Brezis and Kamin [6], Cirstea and Radulescu [7], Chen, Zhou and Ni [8], Crandall, Rabinowitz and Tartar [9], Coclite and Palmieri [10], Dalmasso [11], Goncalves and Santos [12], Iovanov [13], Lair and Shaker [14,15], Lazer and McKenna [16], and references therein. More recently, the system of the type (1.1) has been considered by Ghergu [17] and Zhang [18] where Fi has singular nonlinearities in all variables and by Conti, Merizzi and Terracini [19] for non-singular nonlinearities in all variables. Motivated by these we establish results when the nonlinearities are singular in one of the variables and non-singular in the others; we also consider non-monotonic F1 , F2 . In addition, systems such as (1.1) also have a practical use since reaction–diffusion problems involve multiple interacting objects (Leung [20]). For example, the nonlinear Fi can describe the interplay between two chemicals in a reaction–diffusion process ut − ∆u = F1 (u, v),
vt − ∆v = F2 (u, v) in Ω ,
where no reaction takes place when at least one of the densities vanishes. [19,5]. Of course, it is always of interest to see if results for the single equation can be extended to the system. Our first result addresses the case when Ω is a bounded domain. Theorem 1.1. Let Ω ⊂ RN (N ≥ 2) be a bounded domain in RN with a smooth boundary ∂ Ω . If (F1)–(F3) hold then problem (1.1) has at least one solution (u, v) ∈ C 2 (Ω ) × C 2 (Ω ). For Ω = RN (N ≥ 3), we have Theorem 1.2. If (F1)–(F3) hold with Ω replaced with RN (N ≥ 3) and ai satisfy the condition ∞
rA (r ) dr < ∞
where A (r ) := max (a1 (x) + a2 (x)) ,
(1.3)
|x|=r
0
then problem (1.1) has at least one solution (u, v) ∈ C 2 RN × C 2 RN .
In the next result, we show that condition (1.3) remains nearly necessary in the radially symmetric case for the system (1.1). Theorem 1.3. Assume that Fi (◦, t1 , t2 ) : RN × [0, ∞) × [0, ∞) → (0, ∞) are continuous functions, ∞ possibly singular at the origin. If there exist ε > 0, r0 ≥ 0, and a continuous function B : [r0 , ∞) → (0, ∞) such that r rB (r ) dr = ∞ and for all x ∈ RN with |x| ≥ r0 we have bounded solutions.
0
i=1 ai (x) Fi (x, u, v) ≥ B (r ) for all |(u, v)| ≤ ε , then the system (1.1) admits no radial positive
2
Typical examples of nonlinearities F1 , F2 for our main results are F1 (x, u, v) = uα1 (ε2 + v)α2 (for α1 , α2 < 0 and ε2 > 0) and F2 (x, u, v) = (ε1 + u)α1 v α2 (for α1 , α2 < 0 and ε1 > 0) for which the system (1.1) becomes
−∆u = a1 (x) uα1 (ε2 + v)α2 in Ω , −∆v = a2 (x) (ε1 + u)α1 v α2 in Ω , α
u=v=0
(1.4)
in ∂ Ω . α
α
Then we can easily see that h1 (t1 , t2 ) := t1 1 (ε2 + t2 )α2 , h2 (t1 , t2 ) := (ε1 + t1 )α1 t2 2 , g1 (t1 ) = ε2 2 t1 1 and g2 (t2 ) = +α α ε1 1 t2 2 . Another class of nonlinearities are F1 (x, u, v) = e−α1 u−α2 v (for α1 , α2 > 0) and F2 (x, u, v) = e−β1 u−β2 v (for β1 , β2 > 0). +α
2. Preliminary results Before establishing our main results, we need to develop some auxiliary lemmas. First, we recall an inequality found in [21]. 1/2 Lemma 2.1. For i = 1, 2 let wi ∈ L∞ (Ω ) such that wi > 0 a.e. in Ω , wi ∈ W 1,2 (Ω ), ∆wi ∈ L∞ (Ω ) and w1 = w2 on ∂ Ω . Then
Ω
1/2
1/2
−∆ w 1 1/2
w1
+
∆w2
1/2
w2
(w1 − w2 ) ≥ 0,
if (wi /wj ) ∈ L∞ (Ω ) for i ̸= j, i, j = 1, 2. The next result can be found in the work of Zhang [18, Lemma 3.1], Lee, Shivaji and Ye [22] respectively Alves & Moussaoui [23] (see also Mckenna & Walter [24] for a starting work).
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D.-P. Covei / Applied Mathematics Letters 37 (2014) 118–123
Lemma 2.2. Let Ω ⊂ RN (N ≥ 2) be a bounded domain with a smooth boundary, and let the functions gi := gi (x, t1 , t2 ) : Ω × (0, ∞) × (0, ∞) → R (i= 1, 2) be continuously on their domain. Suppose furthermore that there exist functions ui : Ω → R in C 2 (Ω ) ∩ C Ω (i = 1, 2) such that
−∆ui ≤ gi x, u1 , u2
ui > 0 in Ω ,
in Ω ,
ui = 0 on ∂ Ω
(called a sub-solution); ui : Ω → R in C 2 (Ω ) ∩ C Ω such that
−∆ui ≥ gi (x, u1 , u2 )
in Ω ,
ui > 0 in Ω ,
ui = 0 on ∂ Ω
(called a super-solution) such that ui ≤ ui on Ω . Then, the Dirichlet system in Ω ,
−∆ u i = g i ( x , u 1 , u 2 )
ui = 0 on ∂ Ω , i = 1, 2
(2.1)
has at least one solution (u1 , u2 ) ∈ C (Ω ) ∩ C Ω × C 2 (Ω ) ∩ C Ω with ui ≤ ui ≤ ui on Ω .
2
The following result can be found in [25]. Lemma 2.3. Let Ω be a bounded domain in RN (N ≥ 2) with a smooth boundary ∂ Ω . If a(x) ∈ C 0,α (Ω ) for some α ∈ (0, 1) and a(x) > 0 for all x ∈ Ω then
−∆ω = a (x)
in Ω ,
ω > 0 in Ω , has a unique solution ωa ∈ C 2 (Ω ) ∩ C Ω .
ω (x) = 0 on ∂ Ω
(2.2)
3. Proof of Theorem 1.1 The main point here is the construction of a well-ordered pair of sub and super-solution of (1.1). To do this, let ϕ1i ∈ C (Ω ) ∩ C Ω be the eigenfunction associated with the first eigenvalue λi1 of the problem 2
−∆ϕ = λai (x) ϕ in Ω ,
ϕ > 0 in Ω ,
ϕ = 0 on ∂ Ω .
(3.1)
Using our assumptions on hi and ai (i = 1, 2) of lims→0+ hi (s, s) /s = ∞ and ai ∈ C (Ω ) with ai (x) > 0 for all x ∈ Ω it follows that there exists a parameter δi ∈ (0, ϑi ) such that hi (s, s)
≥ λi1 ,
∀ s ∈ (0, δi ) , i = 1, 2. We want to prove that u, v = c1 ϕ11 , c2 ϕ12 is a sub-solution of (1.1) where ϕ1i > 0 (i = 1, 2) is the eigenfunction associated with λi1 , and ci are positive constants chosen such that δ and δ := min {δ1 , δ2 } , 0 < ci < min 1, 2 max ϕ1i (x) s
Ω
Inspection of the above shows that
−∆u = λ11 a1 (x) c1 ϕ11 (x) ≤ λ11 a1 (x) Σi2=1 ci max ϕ1i (x) Ω 2 i 2 i ≤ a1 (x) h1 Σi=1 ci max ϕ1 (x) , Σi=1 ci max ϕ1 (x) Ω Ω ≤ a1 (x) h1 c1 max ϕ11 , c2 max ϕ12 ≤ a1 (x) h1 c1 ϕ11 (x) , c2 ϕ12 (x) ≤ a1 (x) F1 (x, u, v) Ω
Ω
and moreover −∆v ≤ a2 (x) F2 (x, u, v). Thus we have constructed a sub-solution. Next, we construct a super-solution. Let ωa1 +a2 (x) be the solution to the problem (2.2)
−∆ω = a1 (x) + a2 (x)
in Ω ,
ω > 0 in Ω ,
ω (x) = 0 on ∂ Ω .
We can define u : Ω → [0, ∞) and v : Ω → [0, ∞) implicitly by
ωa1 +a2 (x) =
1 mi
1
mi
u(x)
g i (t ) + 1
0
v(x)
0
t
t g i (t ) + 1
dt
if i = 1
, dt
if i = 2
g i (t ) := t
t2
s ds 0 gi (s)
for i = 1, 2
D.-P. Covei / Applied Mathematics Letters 37 (2014) 118–123
121
where m1 and m2 are constants such that mi
mi max ωa1 +a2 (x) ≤ x∈Ω
t
for i = 1, 2.
g i (t ) + 1
0
It follows that 0 < u ≤ m1 ,
0 < v ≤ m2 ,
u, v ∈ C 2 (Ω ) ∩ C Ω ,
u (x) = v (x) = 0
on ∂ Ω
and
−∆u (x) ≥ a1 (x) g1 (u (x)) ≥ a1 (x) F1 (x, u, v) for all x ∈ Ω , −∆v (x) ≥ a2 (x) g2 (v (x)) ≥ a2 (x) F2 (x, u, v) for all x ∈ Ω . Now we assert that u ≤ u for all x ∈ Ω . Suppose, contrary to the assertion, that
Ωu,u = x ∈ Ω |u (x) > u (x) for all x ∈ Ω ̸= ∅.
(3.2)
Setting w1 := u2 and w2 := u we get, 2
0 ≤
1/2
w
Ωu,u
a1 (x)
≤ Ωu,u
a1 (x)
≤ Ωu,u
∆w 2
+
1/2 1
1/2
−∆w1
w
F1 x, u, v
g1 (u) u
(w1 − w2 )dx =
1/2 2
u
−
g1 (u)
−
u
−∆ u
Ωu,u
u
+
∆u
u
u2 − u
2
dx
u
g 1 ( u)
u2 − u
u2 − u
2
2
dx
dx < 0,
which is impossible. Hence Ωu,u = ∅. Similarly v ≤ v for all x ∈ Ω . On the other hand by u = u = v = v = 0 on ∂ Ω , it follows by Lemma2.1 that u ≤ u in contradicts Ω which the assumption (3.2). Then Lemma 2.2 implies that there exists a function (u, v) ∈ C 2 (Ω ) ∩ C Ω × C 2 (Ω ) ∩ C Ω that solve the system (1.1) with u ≤ u ≤ u on Ω and v ≤ v ≤ v on Ω . 4. Proof of Theorem 1.2 We denote by (un , v n ) the solution obtained in Theorem 1.1 for the system
−∆u = a1 (x) F1 (x, u, v) in Bn −∆v = a2 (x) F2 (x, u, v) in Bn u, v > 0 in Bn , u = v = 0 on ∂ Bn ,
(4.1)
where Bn is the open ball of radius n centered at the origin. Next, we construct an upper bound for this sequence. Following the idea used by Goncalves and Santos [12] we consider u : [0, ∞) → (0, ∞) and v : [0, ∞) → (0, ∞) defined implicitly by
ωA (r ) =
1 M1
1
u(r )
v(r )
M2
t g 1 (t ) + 1
0
t g 2 (t ) + 1
0
dt
,
g i (t ) := t
t2
s ds 0 gi (s)
dt
for i = 1, 2,
where M1 and M2 are constants such that Mi ω (0) ≤
Mi
0
t g i (t ) + 1
for i = 1, 2
and
ωA (r ) :=
1 N −2
∞
σ A(σ )dσ − 0
0
r
ξ 1−N
ξ
σ N −1 A(σ )dσ dξ ,
0
is the unique positive bounded radially symmetric solution of the problem
N lim ω (r ) = 0, −∆ω (r ) = A (r ) on R , r →∞ ∞ 1 ω (0) = σ A(σ )dσ , ω′ (0) = 0 N −2 0
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D.-P. Covei / Applied Mathematics Letters 37 (2014) 118–123
as observed Alves and Holanda in [26]. It follows that u, v ∈ C 2,β RN (for some β ∈ (0, 1)), u (r ) , v (r ) −→ 0 as r → ∞ and
−∆u (r ) ≥ A (r ) g1 (u (r )) ≥ a1 (x) g1 (u (r )) ≥ a1 (x) F1 (x, u (r ) , v (r )) −∆v (r ) ≥ A (r ) g2 (v (r )) ≥ a2 (x) g2 (v (r )) ≥ a2 (x) F2 (x, u (r ) , v (r )) for all x ∈ RN . Moreover, Lemma 2.1 implies 0 < un (x) ≤ u and 0 < v n (x) ≤ v for all x ∈ Bn . A compactness argument, n→∞
n→∞
2 n ∞ n n N proves that there are subsequences of {un }∞ n=1 , {v }n=1 , still denoted N as2 itself, such that u → u and v → v in Cloc (R ) N 2 N and u(x) ≤ u (r ) , v(x) ≤ v (r ) ∀x ∈ R . Finally, (u, v) ∈ C R × C R and (u, v) is a solution of
−∆u = a1 (x) F1 (x, u, v) in RN −∆v = a2 (x) F2 (x, u, v) in RN u, v > 0 in RN , u(x), v(x) → 0 as |x| → +∞,
(4.2)
and the Proof of Theorem 1.2 is finished. 5. Proof of Theorem 1.3 The idea of the proof is to note that if (4.2) has a radial solution, then the function (u, v) satisfies
−(r N −1 u′ (r ))′ = a1 (r ) F1 (r , u(r ), v (r )) , −(r N −1 v ′ (r ))′ = a2 (r ) F2 (r , u(r ), v (r )) , ′ ′ u (r ), v (r ) > 0 for 0 ≤ r < ∞, u (0) = v (0) = 0, u(r ), v (r ) → 0 as r → ∞.
(5.1)
First, we point out that the equations in (5.1) imply u′ (r ) < 0 and v ′ (r ) < 0; i.e. u(r ) and v(r ) are decreasing. Now, summing in (5.1) we obtain r u′′ (r ) + v ′′ (r ) + (N − 1) u′ (r ) + v ′ (r ) ≤ −rB (r ) .
(5.2)
Consequently, by the assumption on the potential function B, the bounded function w defined by w (r ) := u (r ) + v (r ) satisfies r w ′ (r ) − r0 w ′ (r0 ) =
r
t w ′′ (t ) + w ′ (t ) dt =
r0
r
′
t w ′ (t ) dt
r0
r
≤−
tB (t ) dr + (2 − N )
r0
r
w′ (t ) dt → −∞ r0
as r → ∞. Then, there exists a constant M > 0 such that −r w ′ (r ) > M for r > r0 > 0, or equivalently −w ′ (r ) > Mr −1 for r > r0 > 0 and so
w (r0 ) − w (r ) = −
r
w ′ (t ) dt r0
>
r
Mz −1 dz = M ln (r ) − M ln (r0 ) = ∞ for r → ∞ r0
a contradiction (note that w (r0 ) − w (r ) → w (r0 ) as r → ∞). This completes the proof of Theorem 1.3. Acknowledgments The author is grateful to Professor Alan Lair for his numerous suggestions on how to improve the exposition of this work and to the reviewers for the careful reading of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8]
A. Callegari, A. Nashman, Some singular nonlinear equations arising in boundary layer theory, J. Math. Anal. Appl. 64 (1978) 96–105. A.W. Shaker, On singular semilinear elliptic equations, J. Math. Anal. Appl. 173 (1993) 222–228. A. Callegari, A. Nashman, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980) 275–281. R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, Oxford, 1975. J.I. Diaz, Nonlinear partial differential equations and free boundaries, Pitman Research Notes in Mathematics 106 (1985). H. Brezis, S. Kamin, Sublinear elliptic equations in Rn , Manuscripta Math. 74 (1992) 87–106. F. Cirstea, V. Radulescu, Existence and uniqueness of positive solutions to a semilinear elliptic problem in RN , J. Math. Anal. Appl. 229 (1999) 417–425. Goong Chen, Jianxin Zhou, Wei-Ming Ni, Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurcation Chaos 10 (7) (2000) 1565–1612. [9] M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977) 193–222.
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[10] M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989) 1315–1327. [11] R. Dalmasso, Solutions d’ équations elliptiques semi-linéaires singulières, Ann. Mat. Pura Appl. 153 (1988) 191–201. [12] J.V. Goncalves, C.A. Santos, Existence and asymptotic behavior of non-radially symmetric ground states of semilinear singular elliptic equations, Nonlinear Anal. TMA 65 (4) (2006) 719–727. [13] M. Iovanov, Non-existence of ground states for a semilinear elliptic system of Lane–Emden–Fowler type, Carpathian J. Math. 29 (2) (2013) 187–193. [14] A.V. Lair, A.W. Shaker, Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl. 200 (2) (1996) 498–505. [15] A.V. Lair, A.W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl. 211 (2) (1997) 371–385. [16] A.C. Lazer, P.J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991) 721–730. [17] M. Ghergu, Lane–Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295–3318. [18] Zhijun Zhang, Positive solutions of Lane–Emden systems with negative exponents: existence, boundary behavior and uniqueness, Nonlinear Anal. 74 (2011) 5544–5553. [19] M. Conti, L. Merizzi, S. Terracini, On the existence of many solutions for a class of superlinear elliptic systems, J. Differential Equations 167 (2000) 357–387. [20] A. Leung, Nonlinear Systems of Partial Differential Equations—Applications to Life and Physical Sciences, World Scientific Publishing Co. Pte. Ltd., 2009, ISBN-13 978-981-4277-69-3, ISBN-10 981-4277-69-X. [21] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. TMA 10 (1986) 55–64. [22] E.K. Lee, R. Shivaji, J. Ye, Classes of infinite semipositone systems, Proc. Roy. Soc. Edinburgh 139A (2009) 853–865. [23] C.O. Alves, A. Moussaoui, Existence of solution for a class of singular elliptic system with a convection term, arXiv:1307.1014 [math.AP]. [24] P.J. Mckenna, W. Walter, On the dirichlet problem for elliptic systems, Appl. Anal. 21 (3) (1986). [25] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. [26] C.O. Alves, A.R.F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differential Integral Equations 26 (1–2) (2013) 105–118.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Existence and non-existence of solutions for an elliptic system Dragos-Patru Covei ∗ Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st District, Postal Code: 010374, Postal Office: 22, Romania
article
info
abstract
Article history: Received 24 March 2014 Received in revised form 9 June 2014 Accepted 10 June 2014 Available online 21 June 2014
We study the existence of positive solutions for a system of two elliptic equations of the form
−∆u = a1 (x)F1 (x, u, v) −∆v = a2 (x)F2 (x, u, v) u = v = 0 on ∂ Ω
Keywords: Elliptic systems Existence Non-existence
in Ω in Ω
where Ω ⊂ RN (N ≥ 2) is a bounded domain in RN with a smooth boundary ∂ Ω or Ω = RN (N ≥ 3). A non-existence result is obtained for radially symmetric solutions. Our proofs are based primarily on the sub and super-solution method. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study the existence and non-existence of positive solutions for the possibly singular Dirichlet problem for the Lane–Emden–Fowler system
−∆u = a1 (x) F1 (x, u, v) in Ω , −∆v = a2 (x) F2 (x, u, v) in Ω , u = v = 0 in ∂ Ω ,
(1.1)
where Ω ⊂ RN (N ≥ 2) is a bounded domain with a smooth boundary ∂ Ω or Ω = RN (N ≥ 3). (If Ω = RN , then the condition u = v = 0 on ∂ Ω should be understood as u (x), v (x) → 0 as |x| → ∞). Each ai is a positive C 0,α (Ω ) (α ∈ (0, 1)) function, and the function Fi : Ω × (0, ∞) × (0, ∞) → (0, ∞) is continuously on its domain and satisfies the following: (F1) Fi is locally Hölder for each i; g (s) (F2) for every i ∈ {1, 2} there exists a continuous function gi : (0, ∞) → (0, ∞), for which i s is decreasing on (0, ∞) and lims→∞ gi (s) /s := 0, such that Fi (x, t1 , t2 ) ≤ gi (ti ) for all (x, t1 , t2 ) ∈ Ω × (0, ∞) × (0, ∞); (F3) for every i ∈ {1, 2} there exists hi : (0, ϑ1 ) × (0, ϑ2 ) → (0, ∞) continuous non-increasing function such that Fi (x, t1 , t2 ) ≥ hi (t1 , t2 ) for all (x, t1 , t2 ) ∈ Ω × (0, ϑ1 ) × (0, ϑ2 ), for some ϑi ∈ (0, 1) and lims→0 hi (s, s) ∈ (0, ∞]. The scalar case corresponding to the system (1.1)
−∆u = F (x, u) in Ω , ∗
u=0
on ∂ Ω
Tel.: +40 0766224814. E-mail addresses: [email protected], [email protected].
http://dx.doi.org/10.1016/j.aml.2014.06.007 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
(1.2)
D.-P. Covei / Applied Mathematics Letters 37 (2014) 118–123
119
has been studied by several authors and applied to a variety of problems. Examples are boundary layer phenomena for viscous fluids (Callegari–Nashman [1], Shaker [2]), pseudoplastic fluids (Callegari–Nashman [3]), reaction–diffusion processes and chemical heterogeneous catalysts (Aris [4]) and heat conduction in electrically conducting materials (Diaz [5]). Others have studied similar problems with less focus on specific applications such as Brezis and Kamin [6], Cirstea and Radulescu [7], Chen, Zhou and Ni [8], Crandall, Rabinowitz and Tartar [9], Coclite and Palmieri [10], Dalmasso [11], Goncalves and Santos [12], Iovanov [13], Lair and Shaker [14,15], Lazer and McKenna [16], and references therein. More recently, the system of the type (1.1) has been considered by Ghergu [17] and Zhang [18] where Fi has singular nonlinearities in all variables and by Conti, Merizzi and Terracini [19] for non-singular nonlinearities in all variables. Motivated by these we establish results when the nonlinearities are singular in one of the variables and non-singular in the others; we also consider non-monotonic F1 , F2 . In addition, systems such as (1.1) also have a practical use since reaction–diffusion problems involve multiple interacting objects (Leung [20]). For example, the nonlinear Fi can describe the interplay between two chemicals in a reaction–diffusion process ut − ∆u = F1 (u, v),
vt − ∆v = F2 (u, v) in Ω ,
where no reaction takes place when at least one of the densities vanishes. [19,5]. Of course, it is always of interest to see if results for the single equation can be extended to the system. Our first result addresses the case when Ω is a bounded domain. Theorem 1.1. Let Ω ⊂ RN (N ≥ 2) be a bounded domain in RN with a smooth boundary ∂ Ω . If (F1)–(F3) hold then problem (1.1) has at least one solution (u, v) ∈ C 2 (Ω ) × C 2 (Ω ). For Ω = RN (N ≥ 3), we have Theorem 1.2. If (F1)–(F3) hold with Ω replaced with RN (N ≥ 3) and ai satisfy the condition ∞
rA (r ) dr < ∞
where A (r ) := max (a1 (x) + a2 (x)) ,
(1.3)
|x|=r
0
then problem (1.1) has at least one solution (u, v) ∈ C 2 RN × C 2 RN .
In the next result, we show that condition (1.3) remains nearly necessary in the radially symmetric case for the system (1.1). Theorem 1.3. Assume that Fi (◦, t1 , t2 ) : RN × [0, ∞) × [0, ∞) → (0, ∞) are continuous functions, ∞ possibly singular at the origin. If there exist ε > 0, r0 ≥ 0, and a continuous function B : [r0 , ∞) → (0, ∞) such that r rB (r ) dr = ∞ and for all x ∈ RN with |x| ≥ r0 we have bounded solutions.
0
i=1 ai (x) Fi (x, u, v) ≥ B (r ) for all |(u, v)| ≤ ε , then the system (1.1) admits no radial positive
2
Typical examples of nonlinearities F1 , F2 for our main results are F1 (x, u, v) = uα1 (ε2 + v)α2 (for α1 , α2 < 0 and ε2 > 0) and F2 (x, u, v) = (ε1 + u)α1 v α2 (for α1 , α2 < 0 and ε1 > 0) for which the system (1.1) becomes
−∆u = a1 (x) uα1 (ε2 + v)α2 in Ω , −∆v = a2 (x) (ε1 + u)α1 v α2 in Ω , α
u=v=0
(1.4)
in ∂ Ω . α
α
Then we can easily see that h1 (t1 , t2 ) := t1 1 (ε2 + t2 )α2 , h2 (t1 , t2 ) := (ε1 + t1 )α1 t2 2 , g1 (t1 ) = ε2 2 t1 1 and g2 (t2 ) = +α α ε1 1 t2 2 . Another class of nonlinearities are F1 (x, u, v) = e−α1 u−α2 v (for α1 , α2 > 0) and F2 (x, u, v) = e−β1 u−β2 v (for β1 , β2 > 0). +α
2. Preliminary results Before establishing our main results, we need to develop some auxiliary lemmas. First, we recall an inequality found in [21]. 1/2 Lemma 2.1. For i = 1, 2 let wi ∈ L∞ (Ω ) such that wi > 0 a.e. in Ω , wi ∈ W 1,2 (Ω ), ∆wi ∈ L∞ (Ω ) and w1 = w2 on ∂ Ω . Then
Ω
1/2
1/2
−∆ w 1 1/2
w1
+
∆w2
1/2
w2
(w1 − w2 ) ≥ 0,
if (wi /wj ) ∈ L∞ (Ω ) for i ̸= j, i, j = 1, 2. The next result can be found in the work of Zhang [18, Lemma 3.1], Lee, Shivaji and Ye [22] respectively Alves & Moussaoui [23] (see also Mckenna & Walter [24] for a starting work).
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Lemma 2.2. Let Ω ⊂ RN (N ≥ 2) be a bounded domain with a smooth boundary, and let the functions gi := gi (x, t1 , t2 ) : Ω × (0, ∞) × (0, ∞) → R (i= 1, 2) be continuously on their domain. Suppose furthermore that there exist functions ui : Ω → R in C 2 (Ω ) ∩ C Ω (i = 1, 2) such that
−∆ui ≤ gi x, u1 , u2
ui > 0 in Ω ,
in Ω ,
ui = 0 on ∂ Ω
(called a sub-solution); ui : Ω → R in C 2 (Ω ) ∩ C Ω such that
−∆ui ≥ gi (x, u1 , u2 )
in Ω ,
ui > 0 in Ω ,
ui = 0 on ∂ Ω
(called a super-solution) such that ui ≤ ui on Ω . Then, the Dirichlet system in Ω ,
−∆ u i = g i ( x , u 1 , u 2 )
ui = 0 on ∂ Ω , i = 1, 2
(2.1)
has at least one solution (u1 , u2 ) ∈ C (Ω ) ∩ C Ω × C 2 (Ω ) ∩ C Ω with ui ≤ ui ≤ ui on Ω .
2
The following result can be found in [25]. Lemma 2.3. Let Ω be a bounded domain in RN (N ≥ 2) with a smooth boundary ∂ Ω . If a(x) ∈ C 0,α (Ω ) for some α ∈ (0, 1) and a(x) > 0 for all x ∈ Ω then
−∆ω = a (x)
in Ω ,
ω > 0 in Ω , has a unique solution ωa ∈ C 2 (Ω ) ∩ C Ω .
ω (x) = 0 on ∂ Ω
(2.2)
3. Proof of Theorem 1.1 The main point here is the construction of a well-ordered pair of sub and super-solution of (1.1). To do this, let ϕ1i ∈ C (Ω ) ∩ C Ω be the eigenfunction associated with the first eigenvalue λi1 of the problem 2
−∆ϕ = λai (x) ϕ in Ω ,
ϕ > 0 in Ω ,
ϕ = 0 on ∂ Ω .
(3.1)
Using our assumptions on hi and ai (i = 1, 2) of lims→0+ hi (s, s) /s = ∞ and ai ∈ C (Ω ) with ai (x) > 0 for all x ∈ Ω it follows that there exists a parameter δi ∈ (0, ϑi ) such that hi (s, s)
≥ λi1 ,
∀ s ∈ (0, δi ) , i = 1, 2. We want to prove that u, v = c1 ϕ11 , c2 ϕ12 is a sub-solution of (1.1) where ϕ1i > 0 (i = 1, 2) is the eigenfunction associated with λi1 , and ci are positive constants chosen such that δ and δ := min {δ1 , δ2 } , 0 < ci < min 1, 2 max ϕ1i (x) s
Ω
Inspection of the above shows that
−∆u = λ11 a1 (x) c1 ϕ11 (x) ≤ λ11 a1 (x) Σi2=1 ci max ϕ1i (x) Ω 2 i 2 i ≤ a1 (x) h1 Σi=1 ci max ϕ1 (x) , Σi=1 ci max ϕ1 (x) Ω Ω ≤ a1 (x) h1 c1 max ϕ11 , c2 max ϕ12 ≤ a1 (x) h1 c1 ϕ11 (x) , c2 ϕ12 (x) ≤ a1 (x) F1 (x, u, v) Ω
Ω
and moreover −∆v ≤ a2 (x) F2 (x, u, v). Thus we have constructed a sub-solution. Next, we construct a super-solution. Let ωa1 +a2 (x) be the solution to the problem (2.2)
−∆ω = a1 (x) + a2 (x)
in Ω ,
ω > 0 in Ω ,
ω (x) = 0 on ∂ Ω .
We can define u : Ω → [0, ∞) and v : Ω → [0, ∞) implicitly by
ωa1 +a2 (x) =
1 mi
1
mi
u(x)
g i (t ) + 1
0
v(x)
0
t
t g i (t ) + 1
dt
if i = 1
, dt
if i = 2
g i (t ) := t
t2
s ds 0 gi (s)
for i = 1, 2
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where m1 and m2 are constants such that mi
mi max ωa1 +a2 (x) ≤ x∈Ω
t
for i = 1, 2.
g i (t ) + 1
0
It follows that 0 < u ≤ m1 ,
0 < v ≤ m2 ,
u, v ∈ C 2 (Ω ) ∩ C Ω ,
u (x) = v (x) = 0
on ∂ Ω
and
−∆u (x) ≥ a1 (x) g1 (u (x)) ≥ a1 (x) F1 (x, u, v) for all x ∈ Ω , −∆v (x) ≥ a2 (x) g2 (v (x)) ≥ a2 (x) F2 (x, u, v) for all x ∈ Ω . Now we assert that u ≤ u for all x ∈ Ω . Suppose, contrary to the assertion, that
Ωu,u = x ∈ Ω |u (x) > u (x) for all x ∈ Ω ̸= ∅.
(3.2)
Setting w1 := u2 and w2 := u we get, 2
0 ≤
1/2
w
Ωu,u
a1 (x)
≤ Ωu,u
a1 (x)
≤ Ωu,u
∆w 2
+
1/2 1
1/2
−∆w1
w
F1 x, u, v
g1 (u) u
(w1 − w2 )dx =
1/2 2
u
−
g1 (u)
−
u
−∆ u
Ωu,u
u
+
∆u
u
u2 − u
2
dx
u
g 1 ( u)
u2 − u
u2 − u
2
2
dx
dx < 0,
which is impossible. Hence Ωu,u = ∅. Similarly v ≤ v for all x ∈ Ω . On the other hand by u = u = v = v = 0 on ∂ Ω , it follows by Lemma2.1 that u ≤ u in contradicts Ω which the assumption (3.2). Then Lemma 2.2 implies that there exists a function (u, v) ∈ C 2 (Ω ) ∩ C Ω × C 2 (Ω ) ∩ C Ω that solve the system (1.1) with u ≤ u ≤ u on Ω and v ≤ v ≤ v on Ω . 4. Proof of Theorem 1.2 We denote by (un , v n ) the solution obtained in Theorem 1.1 for the system
−∆u = a1 (x) F1 (x, u, v) in Bn −∆v = a2 (x) F2 (x, u, v) in Bn u, v > 0 in Bn , u = v = 0 on ∂ Bn ,
(4.1)
where Bn is the open ball of radius n centered at the origin. Next, we construct an upper bound for this sequence. Following the idea used by Goncalves and Santos [12] we consider u : [0, ∞) → (0, ∞) and v : [0, ∞) → (0, ∞) defined implicitly by
ωA (r ) =
1 M1
1
u(r )
v(r )
M2
t g 1 (t ) + 1
0
t g 2 (t ) + 1
0
dt
,
g i (t ) := t
t2
s ds 0 gi (s)
dt
for i = 1, 2,
where M1 and M2 are constants such that Mi ω (0) ≤
Mi
0
t g i (t ) + 1
for i = 1, 2
and
ωA (r ) :=
1 N −2
∞
σ A(σ )dσ − 0
0
r
ξ 1−N
ξ
σ N −1 A(σ )dσ dξ ,
0
is the unique positive bounded radially symmetric solution of the problem
N lim ω (r ) = 0, −∆ω (r ) = A (r ) on R , r →∞ ∞ 1 ω (0) = σ A(σ )dσ , ω′ (0) = 0 N −2 0
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as observed Alves and Holanda in [26]. It follows that u, v ∈ C 2,β RN (for some β ∈ (0, 1)), u (r ) , v (r ) −→ 0 as r → ∞ and
−∆u (r ) ≥ A (r ) g1 (u (r )) ≥ a1 (x) g1 (u (r )) ≥ a1 (x) F1 (x, u (r ) , v (r )) −∆v (r ) ≥ A (r ) g2 (v (r )) ≥ a2 (x) g2 (v (r )) ≥ a2 (x) F2 (x, u (r ) , v (r )) for all x ∈ RN . Moreover, Lemma 2.1 implies 0 < un (x) ≤ u and 0 < v n (x) ≤ v for all x ∈ Bn . A compactness argument, n→∞
n→∞
2 n ∞ n n N proves that there are subsequences of {un }∞ n=1 , {v }n=1 , still denoted N as2 itself, such that u → u and v → v in Cloc (R ) N 2 N and u(x) ≤ u (r ) , v(x) ≤ v (r ) ∀x ∈ R . Finally, (u, v) ∈ C R × C R and (u, v) is a solution of
−∆u = a1 (x) F1 (x, u, v) in RN −∆v = a2 (x) F2 (x, u, v) in RN u, v > 0 in RN , u(x), v(x) → 0 as |x| → +∞,
(4.2)
and the Proof of Theorem 1.2 is finished. 5. Proof of Theorem 1.3 The idea of the proof is to note that if (4.2) has a radial solution, then the function (u, v) satisfies
−(r N −1 u′ (r ))′ = a1 (r ) F1 (r , u(r ), v (r )) , −(r N −1 v ′ (r ))′ = a2 (r ) F2 (r , u(r ), v (r )) , ′ ′ u (r ), v (r ) > 0 for 0 ≤ r < ∞, u (0) = v (0) = 0, u(r ), v (r ) → 0 as r → ∞.
(5.1)
First, we point out that the equations in (5.1) imply u′ (r ) < 0 and v ′ (r ) < 0; i.e. u(r ) and v(r ) are decreasing. Now, summing in (5.1) we obtain r u′′ (r ) + v ′′ (r ) + (N − 1) u′ (r ) + v ′ (r ) ≤ −rB (r ) .
(5.2)
Consequently, by the assumption on the potential function B, the bounded function w defined by w (r ) := u (r ) + v (r ) satisfies r w ′ (r ) − r0 w ′ (r0 ) =
r
t w ′′ (t ) + w ′ (t ) dt =
r0
r
′
t w ′ (t ) dt
r0
r
≤−
tB (t ) dr + (2 − N )
r0
r
w′ (t ) dt → −∞ r0
as r → ∞. Then, there exists a constant M > 0 such that −r w ′ (r ) > M for r > r0 > 0, or equivalently −w ′ (r ) > Mr −1 for r > r0 > 0 and so
w (r0 ) − w (r ) = −
r
w ′ (t ) dt r0
>
r
Mz −1 dz = M ln (r ) − M ln (r0 ) = ∞ for r → ∞ r0
a contradiction (note that w (r0 ) − w (r ) → w (r0 ) as r → ∞). This completes the proof of Theorem 1.3. Acknowledgments The author is grateful to Professor Alan Lair for his numerous suggestions on how to improve the exposition of this work and to the reviewers for the careful reading of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8]
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