Existence results for some nonlinear elliptic systems

Nonlinear Analysis 71 (2009) 2840–2846

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Existence results for some nonlinear elliptic systems Jihui Zhang a,∗ , Zhitao Zhang b a

Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, People’s Republic of China

b

Academy of Mathematics and Systems Sciences, Institute of Mathematics, The Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

article

abstract

info

Article history: Received 9 April 2008 Accepted 20 January 2009

The paper is concerned with some nonlinear elliptic systems. By using variational methods, we obtain the existence of weak solutions for the semilinear elliptic system and the quasilinear elliptic system. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Weak solution Nonlinear elliptic system Variational methods

1. Introduction In this paper we consider the following system of semilinear elliptic equations:

( −∆u = f (x, u, v) −∆v = g (x, u, v) u=v=0

in Ω , in Ω , on ∂ Ω ,

(1.1)

where Ω is a bounded smooth open set in Rn , ∆ is the elliptic Laplacian on Rn , f and g are Carathéodory functions. We also consider the following system of quasilinear elliptic equations:



−div h1 |∇ u|2
∇ u
= f (x, u, v) −div h2 |∇v|2 ∇v = g (x, u, v)  u=v=0

in Ω , in Ω , on ∂ Ω ,

(1.2)

where h1 and h2 ∈ C (R, R). In recent years, the existence of solutions for elliptic systems have been widely studied. Costa and Magalhes [1,2] consider subquadratic perturbations of semilinear elliptic systems by minimax methods. Cao and Tang [3] consider a class of superlinear elliptic systems by variational methods. Bartsch and Clapp [4] study an elliptic system by critical point theorems. In [5] it was proved multiplicity results for elliptic systems by using an abstract linking theorem. In [6] by using Morse theory, they discuss an asymptotically linear cooperative elliptic system at resonance. For other papers see [7–17] and the references therein. It is the purpose of this paper to use variational methods for semilinear elliptic system and quasilinear elliptic system. We show the existence of solutions of elliptic system (1.1) and elliptic system (1.2) for nonlinearities f and g under another hypothesis. For (u, v) ∈ R2 , denote |(u, v)|2 = |u|2 + |v|2 . We assume that W : Ω × R2 → R is of C 1 class such that (f , g ) = ( ∂∂Wu , ∂∂vW ), f and g are Carathéodory functions satisfying the following growth conditions:

∗ Corresponding address: School of Mathematics and Computer Sciences, Nanjing Normal University, 122 Ninghai Road, 210097Nanjing, Jiangsu, People’s Republic of China. Tel.: +86 025 86272816. E-mail addresses: [email protected] (J. Zhang), [email protected] (Z. Zhang). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.158

J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

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(i) There exist ai ≥ 0, (i = 1, 2), f1 and g1 ∈ Lqi (Ω ) such that

|f (x, u, v)| ≤ a1 |(u, v)|α1 + f1 (x), α2

|g (x, u, v)| ≤ a2 |(u, v)|

(1.3)

+ g1 (x),

(1.4)

for a.e. x ∈ Ω and for all u, v ∈ R, where 1 ≤ αi < (n + 2)/(n − 2) if n ≥ 3 and 1 ≤ αi < +∞ if n = 1, 2, pi = αi + 1 and 1/pi + 1/qi = 1, (i = 1, 2); (ii) There exists λ ∈ L∞ (Ω ) such that lim sup

2W (x, u, v) u2 + v 2

|(u,v)|→∞

≤ λ(x),

(1.5)

uniformly for a.e. x ∈ Ω ; (iii) Let h1 and h2 ∈ C (R, R). We assume that h1 and h2 are the continuous and nondecreasing functions satisfying the following growth conditions: There exist α1 , α2 , β1 and β2 ∈ R such that 0 < α1 ≤ h1 (t ) ≤ β1 ,

(1.6)

0 < α2 ≤ h2 (t ) ≤ β2 .

(1.7)

Under the above assumptions, when the nonlinearities f and g are sublinear at ∞, and satisfy the weak condition λ(x) ≤ λ1 on Ω and λ(x) < λ1 on a subset of Ω with positive measure or λ(x) ≤ min(α1 , α1 )λ1 on Ω and λ(x) < min(α1 , α2 )λ1 on a subset of Ω with positive measure. We shall give the existence of weak solutions, by means of variational methods, for the semilinear elliptic system (1.1) and the quasilinear elliptic system (1.2). The existence of a weak solution (u, v) ∈ H01 (Ω ) × H01 (Ω ) is proved in each of the cases (1.1) and (1.2) through minimization of the pertinent functional. The main results of this paper are the following: Theorem 1.1. Assume that (i) and (ii) hold. Moreover, suppose that λ(x) ≤ λ1 on Ω and λ(x) < λ1 on a subset of Ω with positive measure. Then system (1.1) has at least one weak solution. Theorem 1.2. Assume that (i)–(iii) hold. Moreover, suppose that λ(x) ≤ min(α1 , α1 )λ1 on Ω and λ(x) < min(α1 , α2 )λ1 on a subset of Ω with positive measure. Then system (1.2) has at least one weak solution. Remark 1.1. Theorem 1.2 is new and includes an interesting interplay between the behavior of the functions h1 (s), h2 (s) and the potential function W (x, u, v). On the other hand, Theorem 1.1 is an extension to 2 × 2 systems of a result by Mawhin, Ward and Willem [18], as well as of another paper by Costa and Oliveira [19] dealing with a similar subquadratic situation between two consecutive eigenvalues, Cao and Li [20] discuss a variational problem. This paper is organized as follows. In Section 2, we give some notations and review some relevant lemmas. The main results are proved in Section 3. 2. Notations and preliminary lemmas Given a bounded smooth open set Ω ⊂ Rn . Here, let us consider the Hilbert space H = H01 (Ω ) × H01 (Ω ). Let H 0 be dual of H, h , i the duality pairing between H 0 and H, and h , iL2 the inner product in L2 (Ω ). The norm on H is given by

k(u, v)k = kuk2 + kvk2


1/2

(2.1)

and the norm on L2 (Ω ) × L2 (Ω ) is given by

k(u, v)kL2 = kuk2L2 + kvk2L2


1/2

,

(2.2)

where k k and k kL2 are the norm on H01 (Ω ) and L2 (Ω ) respectively, that is,

Z k uk = Ω

1/2 |∇ u|2 dx ,

Z kukL2 =

|u|2 Ω

1/2

.

Let λk (k = 1, 2, . . .) denote the eigenvalues and ϕk (k = 1, 2, . . .) the corresponding eigenfunctions, suitably normalized with respect to L2 (Ω ) inner product, of the eigenvalue problem



∆u + λ u = 0 u=0

in Ω , on ∂ Ω ,

(2.3)

where each eigenvalue λk is repeated as often as multiplicity recall that 0 < λ1 < λ2 ≤ λ3 ≤ · · · , λk → ∞, and that ϕ1 (x) > 0 for x ∈ Ω .

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J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

The set of functions {ϕk } is an orthogonal base for H01 (Ω ), thus we may denote an element u of H01 (Ω ) as u=

∞ X

∞ X

ak ϕk ,

k=1

a2k < ∞.

(2.4)

k=1

Let us define the mappings J1 (u, v) =

Z

1 2

[|∇ u|2 + |∇v|2 ]dx

Ω

and J10 : H → H 0 by

hJ10 (u, v), (ξ , ζ )i =

Z Ω

[∇ u∇ξ + ∇v∇ζ ]dx,

(2.5)

for any (u, v), (ξ , ζ ) ∈ H. Let us define the mappings h(u, v) = J2 (u, v) =

1

u

Z

2

h1 (s)ds + 0

Z Ω

v

Z



h2 (s)ds ,

0

[h(|∇ u|2 , |∇v|2 )]dx

and J20 : H → H 0 by

hJ20 (u, v), (ξ , ζ )i =

Z

 Ω

h1 (|∇ u|2 )∇ u∇ξ + h2 (|∇v|2 )∇v∇ζ dx,



(2.6)

for any (u, v), (ξ , ζ ) ∈ H. Let us define the mapping W 0 : H → H 0 by

hW 0 (u, v), (ξ , ζ )i =

Z Ω

[f (x, u, v)ξ + g (x, u, v)ζ ]dx,

(2.7)

for any (u, v), (ξ , ζ ) ∈ H. Definition 2.1. We say that (u, v) ∈ H is a weak solution of the elliptic system (1.1) if the identity

hJ10 (u, v), (ξ , ζ )i = hW 0 (u, v), (ξ , ζ )i

(2.8)

holds for any (ξ , ζ ) ∈ H. Definition 2.2. We say that (u, v) ∈ H is a weak solution of the elliptic system (1.2) if the identity

hJ20 (u, v), (ξ , ζ )i = hW 0 (u, v), (ξ , ζ )i

(2.9)

holds for any (ξ , ζ ) ∈ H. We need the following lemmas: Lemma 2.1. Let λ ∈ L∞ (Ω ), we assume that λ(x) ≤ λ1 on Ω and λ(x) < λ1 on a subset of Ω with positive measure. Then there exists an ε > 0 such that

Z

 Ω

 |∇ u|2 + |∇v|2 − λ(x)(u2 + v 2 ) dx ≥ 2ε(kuk2 + kvk2 ),

(2.10)

for any (u, v) ∈ H. Remark 2.1. For proofs of Theorems 1.1 and 1.2, Lemma 2.1 was used in a crucial way. Indeed, it was precisely the kind of subquadratic hypothesis pointed above that caused the underlying functional to be coercive in the paper by Mawhin,Ward and Willem [18]. At the same time, the proof of Lemma 2.1 is quite different from the original proof by Mawhin, Ward and Willem [18].

J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

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Proof. First of all it is clear to see that:

kuk2 + kvk2 ≥ λ1 (kuk2L2 + kuk2L2 ),

(2.11)

for any (u, v) ∈ H. If (2.10) is false, then there exists a sequence {(un , vn )} ⊂ H, such that k(un , vn )k = 1 and

Z



lim

n→∞

Ω

 |∇ un |2 + |∇vn |2 − λ(x)(u2n + vn2 ) dx = 0,

(2.12)

passing to subsequence, we can suppose that un → u, vn → v weakly in H01 (Ω ), strongly in L2 (Ω ) and

Z lim

n→∞

(|∇ un | + |∇vn | )dx = 2

Ω

2

Z Ω

λ(x)(u2 + v 2 )dx.

(2.13)

Let

(un , vn ) =

∞ X

(n)

ak ϕk ,

∞ X

k=1

!

(n)

bk ϕk

,

(u, v) =

k=1

∞ X

ak ϕk ,

k=1

∞ X

! bk ϕk

,

(2.14)

k=1

from (2.11), we get

Z

Z (|∇ un |2 + |∇vn |2 )dx − λ1 (u2n + vn2 )dx ZΩ Z Ω 2 2 ≤ (|∇ un | + |∇vn | )dx − λ(x)(u2n + vn2 )dx,

0 ≤

Ω

Ω

using (2.12), it implies that

Z lim

n→∞

Ω

(|∇ un |2 + |∇vn |2 )dx − λ1

Z Ω

 (u2n + vn2 )dx = 0.

(2.15)

Inserting (2.14) into (2.15), we obtain lim

∞ X

n→∞



(n)

(n)

[λk − λ1 ] (ak )2 + (bk )2



= 0.

k=2

Since 0 ≤

∞   X (a(kn) )2 + (b(kn) )2 k=2

≤

1

∞ X

λ2 − λ1

k=2

  (λk − λ1 ) (a(kn) )2 + (b(kn) )2 ,

by (2.16), we have lim

n→∞

∞ X (a(kn) )2 = 0,

lim

n→∞

k=2

therefore it follows that ∞ X

a2k = kuk2L2 = lim kun k2L2 n→∞

k=1

= a21 + lim

n→∞

∞ X (a(kn) )2 k=2

= a21 and ∞ X

b2k = kvk2L2 = lim kvn k2L2 n→∞

k=1

= b21 + lim

n→∞

= b21

∞ X k=2

(b(kn) )2

∞ X (b(kn) )2 = 0, k =2

(2.16)

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J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

which implies that ak = 0, bk = 0 (k ≥ 2), then u = a1 ϕ1 , v = b1 ϕ1 , a1 6= 0 or b1 6= 0. By combining (2.13) and (2.15), we get

Z Ω

[λ1 − λ(x)] u2 dx = 0

(2.17)

[λ1 − λ(x)] v 2 dx = 0.

(2.18)

and

Z Ω

Hence u ≡ 0 and v ≡ 0 on a set of positive measure, then ϕ1 ≡ 0 on a set of positive measure. Moreover, we notice that ∆ − λ1 is a real analytic hypoellipticity differential operator (cf. [21]), then ϕ1 is a real analytic function in Ω . By the unique continuation theorem, we obtain that ϕ1 ≡ 0 in Ω . This is a contradiction because kϕ1 kL2 = 1. This completes the proof of Lemma 2.1.  Lemma 2.2. Let W 0 be the mapping given by (2.7), then W 0 is a completely continuous mapping. Proof. By (i) and embedding the mapping I : H → Lp1 (Ω ) × Lp2 (Ω )(pi = αi + 1),

(i = 1, 2),

I (u, v) = (u, v)

is completely continuous. From [22,23], the mapping J : Lp1 (Ω ) × Lp2 (Ω ) → Lq1 (Ω ) × Lq2 (Ω )((1/pi ) + (1/qi ) = 1),

(i = 1, 2),

J (u, v) = (f (x, u, v), g (x, u, v)) is continuous and the mapping K : Lq1 (Ω ) × Lq2 (Ω ) → H 0 ,

K (φ, ψ) = (φ, ψ)

is also continuous. Let W 0 = KJI. Then we know that W 0 is a completely continuous mapping.



3. Proofs of main results In this section we give the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. Let J1 (u, v) = E (u, v) = J1 (u, v) −

Z Ω

1 2

R

Ω

[|∇ u|2 + |∇v|2 ]dx as in Section 2. Let the energy E : H → R be given by

W (x, u, v)dx,

(3.1)

for any (u, v) ∈ H. Then a weak solution of system (1.1) is a critical point of E (u, v) in H. At first, using Lemma 2.1, we get that there exists an ε > 0 such that

Z Ω

(|∇ u|2 + |∇v|2 )dx −

Z Ω

λ(x)(u2 + v 2 )dx ≥ 2ε(kuk2 + kuk2 ),

(3.2)

for any (u, v) ∈ H, by assumptions (i) and (ii), we get that there exists a function ω1 ∈ L1 (Ω ) such that W (x, u, v) ≤ (λ(x) + λ1 ε) (u2 + v 2 )/2 + ω1 (x),

(3.3)

for any (u, v) ∈ H, where ε is given by Lemma 2.1. Thus, by combining (3.1), (3.2) and (3.3), it follows that E (u, v) ≥

1



Z



Z

  hJ1 (u, v), (u, v)i − λ(x)(u + v )dx − λ1 ε(u2 + v 2 )/2 + ω1 (x) dx Ω Ω Z  Z Z   1 2 2 2 2 = (|∇ u| + |∇v| )dx − λ(x)(u + v )dx − λ1 ε(u2 + v 2 )/2 + ω1 (x) dx, 0

2

2 2

Ω

Ω

for any (u, v) ∈ H. Using (3.2) and (2.11) we obtain E (u, v) ≥

2

ε 2

k(u, v)k2 −

Z Ω

ω1 (x)dx,

Ω

J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

ˆ (u, v) = for any (u, v) ∈ H, thus E is coercive on H. Next, let W

R

Ω

2845

ˆ (u, v), by (i) W (x, u, v)dx, then E (u, v) = J1 (u, v) − W

ˆ is weakly continuous. Since J1 is weakly lower semicontinuous, thus E is weakly lower semicontinuous. we know that W Then E has a minimum at some point (u, v) ∈ H. Moreover, by (i) and Lemma 2.2, E is continuously differentiable on H and hE (u, v), (ξ , ζ )i = 0

Z Ω 0

[∇ u∇ξ + ∇v∇ζ − f (x, u, v)ξ − g (x, u, v)ζ ] v dx

= hJ1 (u, v), (ξ , ζ )i − hW 0 (u, v), (ξ , ζ )i,

(3.4)

for any (ξ , ζ ) ∈ H. Thus E 0 (u, v) = 0, this implies that

hJ10 (u, v), (ξ , ζ )i = hW 0 (u, v), (ξ , ζ )i,

(3.5)

for any (ξ , ζ ) ∈ H. Therefore (u, v) is a weak solution of system (1.1). This completes the proof of Theorem 1.1. Proof of Theorem 1.2. Let J2 (u, v) = E (u, v) = J2 (u, v) −

Z Ω

R

Ω



[h(|∇ u| , |∇v| )]dx as in Section 2, and let the energy E : H → R be given by 2

2

W (x, u, v)dx,

(3.6)

ˆ (u, v) = for any (u, v) ∈ H. Then a weak solution of system (1.2) is a critical point of E (u, v) in H. let W

R

Ω

W (x, u, v)dx, then

ˆ . It is easy to check that J2 is a convex and lower semicontinuous, hence J2 is weakly lower semicontinuous, by E = J2 − W ˆ is weakly continuous. Thus E is weakly lower semicontinuous. By assumptions (i), (iii) and Lemma 2.1, (i) we know that W there exists an ε > 0 such that Z

h(|∇ u|2 , |∇v|2 ) − λ(x)(u2 + v 2 )/2 dx ≥

 Ω



1

Z


α1 |∇ u|2 + α2 |∇v|2 − λ(x)(u2 + v 2 ) dx

2 Ω Z   1 ≥ min(α1 , α2 )(|∇ u|2 + |∇v|2 ) − λ(x)(u2 + v 2 ) dx 2 Ω Z   min(α1 , α2 ) = |∇ u|2 + |∇v|2 − λ(x)(u2 + v 2 )/ min(α1 , α2 ) dx 2 Ω

≥ min(α1 , α2 )ε(kuk2 + kvk2 ), for any (u, v) ∈ H, thus we get J2 (u, v) −

1 2

Z Ω

λ(x)(u2 + v 2 )dx ≥ min(α1 , α2 )ε(kuk2 + kvk2 ),

(3.7)

for any (u, v) ∈ H. By assumptions (i) and (ii), there exists a function ω2 ∈ L1 (Ω ) such that W (x, u, v) ≤ (λ(x) + λ1 min(α1 , α2 )ε) (u2 + v 2 )/2 + ω2 (x),

(3.8)

for any (u, v) ∈ H. Thus, by combining (3.7) and (3.8), it follows that E (u, v) ≥ J2 (u, v) −

1

Z

2

Ω

λ(x)(u2 + v 2 )dx −

Z 

1

Ω

2

 λ1 min(α1 , α2 )ε(u2 + v 2 ) + ω2 (x) dx,

for any (u, v) ∈ H. Using (2.11) and (3.7), we obtain E (u, v) ≥

ε 2

min(α1 , α2 )(kuk + kvk ) − 2

2

Z Ω

ω2 (x)dx,

for any (u, v) ∈ H, thus E is coercive on H. By (i), (ii), (iii) and Lemma 2.2, E is continuously differentiable on H and

hE 0 (u, v), (ξ , ζ )i =

Z

h1 (|∇ u|2 )∇ u∇ξ + h2 (|∇v|2 )∇v∇ζ − f (x, u, v)ξ − g (x, u, v)ζ v dx

 Ω



= hJ20 (u, v), (ξ , ζ )i − hW 0 (u, v), (ξ , ζ )i, for any (u, v) ∈ H. Therefore E has a minimum at some point (u, v) ∈ H and E 0 (u, v) = 0. Thus, this implies that

hJ20 (u, v), (ξ , ζ )i = hW 0 (u, v), (ξ , ζ )i, for any (ξ , ζ ) ∈ H, that is, (u, v) is a weak solution of system (1.2). This completes the proof of Theorem 1.2.

(3.9) 

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J. Zhang, Z. Zhang / Nonlinear Analysis 71 (2009) 2840–2846

Acknowledgements The authors are grateful to the referee for his/her useful suggestions and comments which have improved the writing of the paper. This work was carried out during the first author’s visit to the Academy of Mathematics and Systems Sciences (AMSS), Academia Sinica. He would like to thank Prof. Z. Zhang and the members of AMSS very much for their invitation and hospitality. The authors also would like to thank Prof. D. Cao, Prof. Y. Ding, Prof. S. Li and Prof. Z. Zhang for their help and many valuable discussions. This research was supported by the NNSF of China (10871096), Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education and NSF of Education Committee of Jiangsu Province. The second author was supported by NNSF of China (10671195). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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