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Existence of weak solutions for a class of (p,q) -Laplacian systems on resonance
Linear Algebra Letters and its 50 Applications 466 (2015) 102–116 Applied Mathematics (2015) 29–36
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LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue of Jacobi matrixsystems Existence of weak solutions for a problem class of (p, q)-Laplacian on resonance✩ with mixed data Zeng-Qi Ou
Ying Wei 1
School of Mathematics Department and Statistics, Southwest University, People’sand Republic of China of Mathematics, Nanjing Chongqing University 400715, of Aeronautics Astronautics, Nanjing 210016, PR China
article
info a r t i c l e
abstract i n f o
a b s t r a c t
Article history: The existences of weak solutions for a class of (p, q)-Laplacian systems on resonance Received 21 April 2015 Article history: In certain this paper, the inverse eigenvalueconditions problem of are obtained under the Landesman–Lazer-type byreconstructing using G-link 16 January 2014 Received in revised formReceived 1 June 2015 a Jacobi matrix (1999). from its eigenvalues, leading answer principal Theorem due to Dr´ abek and Robinson Our results giveits a positive to Accepted 20 September 2014 Accepted 2 June 2015 submatrix and(2004). part of the eigenvalues of its submatrix an open problem of Zographopoulos online 22 October 2014 Available online 10 JuneAvailable 2015 is considered. The necessary and sufficient conditions for © 2015 Elsevier Ltd. All rights reserved. Submitted by Y. Wei
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
Keywords: MSC: Elliptic systems 15A18 The Landesman–Lazer-type 15A57 conditions Critical point Keywords: G-linking Theorem Jacobi matrix Eigenvalue Inverse problem Submatrix
1. Introduction and main results Consider the existence of weak solutions for the following elliptic systems p−2 α β −∆p u = λ1 a(x)|u| u + λ1 b(x)|u| |v| v + g1 (x, u) − h1 (x) −∆q v = λ1 c(x)|v|q−2 v + λ1 b(x)|u|α |v|β u + g2 (x, v) − h2 (x) u = v = 0
in Ω , in Ω , on ∂Ω ,
(1)
where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary, ∆p u = div(|∇u|p−2 ∇u) is the pLaplacian operator. 1 < p, q < N and α ≥ 0, β ≥ 0 satisfying α+1
+
β+1
= 1.
p q E-mail address: [email protected]. 1 Tel.: +86 13914485239. The nonlinearities g1 , g2 ∈ C(Ω × R, R) and h1 ∈ Lp/(p−1) (Ω ), h2 a, b, c ∈ C(Ω ) ∩ L∞http://dx.doi.org/10.1016/j.laa.2014.09.031 (Ω ) satisfy one of the following conditions: 0024-3795/© 2014 Published by Elsevier Inc.
(2) ∈ Lq/(q−1) (Ω ). The coefficient functions
✩ Supported by National Natural Science Foundation of China (No. 11471267) and the Fundamental Research Funds for the Central Universities (No. XDJK2011C039), the Postdoctoral Research Foundation of Chongqing (No. Xm201319). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.aml.2015.06.004 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
(a1) a+ ̸= 0, where a+ (x) := max{a(x), 0}; (a2) c+ = ̸ 0; (a3) a = c = 0 and b+ ̸= 0. 1 Let W01,p (Ω ) be the Sobolev space with norm ∥u∥p = Ω |∇u|p dx p for any u ∈ W01,p (Ω ). From the p , the embedding W01,p (Ω ) ↩→ Lτ (Ω ) is Sobolev and Rellich embedding theorems, for any 1 ≤ τ < NN−p continuous and compact, and there is a positive constant C = C(τ, p, N ) such that |u|τ ≤ C∥u∥p ,
for all u ∈ W01,p (Ω ),
(3)
where | · |τ denotes the norm of Lτ (Ω ). In the sequel, C also denotes a positive constant related to the above embedding. Let H = W01,p (Ω ) × W01,q (Ω ) be Banach space with the norm ∥(u, v)∥ = ∥u∥p + ∥v∥q for any (u, v) ∈ H. Let us consider the following nonlinear eigenvalue problem with weights p−2 α β −∆p u = λa(x)|u| u + λb(x)|u| |v| v in Ω , q−2 α β (4) −∆q v = λc(x)|v| v + λb(x)|u| |v| u in Ω , u = v = 0 on ∂Ω . Define the functionals φ, ϕ : H → R as α+1 β+1 φ(u, v) = |∇u|p dx + |∇v|q dx, p q Ω Ω β+1 α+1 p a(x)|u| dx + c(x)|v|q dx + b(x)|u|α |v|β uvdx, ϕ(u, v) = p q Ω Ω Ω and the manifold Σ = {(u, v) ∈ H : ψ(u, v) = 1}. We say that φ(u, v), ϕ(u, v) are (p, q)-homogeneous, namely φ(t1/p u, t1/q v) = tφ(u, v),
ϕ(t1/p u, t1/q v) = tϕ(u, v)
for any t > 0 and (u, v) ∈ H,
and if one of (a1)–(a3) is satisfied, Σ is a symmetric nonempty manifold in H. By a standard argument (the similar proof seen in [1,2]), problem (4) has a sequence of eigenvalues with the variational characterization λk = inf
sup φ(u, v),
F ∈Σk (u,v)∈F
where Σk = {F ⊂ Σ : there is an odd, continuous and surjective γ : S k−1 → F } and S k−1 denotes the unit sphere in Rk . On the other hand, let α+1 β+1 λ′1 = inf |∇u|p dx + |∇v|q dx, p q (u,v)∈Σ Ω Ω then λ′1 is also the first eigenvalue of (4), and from the definition of λ1 , it is easy to see that λ1 = λ′1 . Moreover, λ1 is a simple, isolated and positive principal eigenvalue of (4) and corresponds to a positive normalized eigenvalue (µ0 , ν0 ), that is ∥µ0 ∥p + ∥ν0 ∥q = 1, hence the set of all eigenfunctions corresponding to λ1 is E1 := {(t1/p µ0 , t1/q ν0 ) : t ≥ 0} ∪ {(−t1/p µ0 , −t1/q ν0 ) : t ≥ 0}. Since φ(u, v), ϕ(u, v) are (p, q)-homogeneous, there exists a positive constant t0 such that 1/p
1/q
∥t0 µ0 ∥pp + ∥t0 ν0 ∥qq = 1. 1/p
1/q
Let µ1 = t0 µ0 , ν1 = t0 ν0 , then {(t1/p µ0 , t1/q ν0 ) : t ≥ 0}∪{(−t1/p µ0 , −t1/q ν0 ) : t ≥ 0} = {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}, In the following, let E1 := {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}.
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
The set E1 is not a one-dimensional linear subspace of H and the corresponding orthogonal decomposition on H does not hold with respect to the first eigenvalue λ1 . The Landesman–Lazer-type conditions were introduced by Landesman and Lazer in [3] and have been widely used and extended (see [1,2,4–10] and the reference therein). In [1], if p = q ≥ 2 and the certain Landesman–Lazer-type conditions hold, Zographopoulos proved that problem (1) has at least one weak solution, but he does not know whether the similar result holds for p ̸= q. In this paper, we will give out a positive answer. We first state some auxiliary conditions. (F1) There is h ∈ C(Ω , R+ ) such that |g1 (x, t)| ≤ h(x)
and |g2 (x, t)| ≤ h(x) ∀ (x, t) ∈ Ω × R;
(F2) There exist functions g1+ , g1− ∈ C(Ω , R) such that g1+ (x) = lim g1 (x, t),
g1− (x) = lim g1 (x, t)
t→+∞
t→−∞
uniformly for all x ∈ Ω ;
(F3) There exist functions g2+ , g2− ∈ C(Ω , R) such that g2+ (x) = lim g2 (x, t),
g2− (x) = lim g2 (x, t)
t→+∞
t→−∞
uniformly for all x ∈ Ω .
The Landesman–Lazer-type conditions for problem (1) are assumed either (α + 1) g1− µ1 dx + (β + 1) g2− ν1 dx < (α + 1) h1 µ1 dx + (β + 1) h2 ν1 dx Ω
Ω
Ω
< (α + 1) Ω
Ω
g1+ µ1 dx + (β + 1)
Ω
g2+ ν1 dx;
(5)
or (α + 1) Ω
g1− µ1 dx + (β + 1)
Ω
g2− ν1 dx > (α + 1)
h1 µ1 dx + (β + 1)
Ω
> (α + 1) Ω
h2 ν1 dx Ω
g1+ µ1 dx + (β + 1)
Ω
g2+ ν1 dx.
(6)
The main results of this paper are the following theorems. Theorem 1. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold, and g1 , g2 satisfy (F1), (F2), (F3) and (5). If 1 < p < q, assume that the following inequalities hold: − + h1 µ1 dx − g1 µ1 dx < 0 and g1 µ1 dx − h1 µ1 dx < 0, (7) Ω
Ω
Ω
Ω
then problem (1) has at least one solution. In the other case 1 < q < p, we have the following result. Theorem 2. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold, and g1 , g2 satisfy (F1), (F2), (F3) and (5). If 1 < q < p, assume that the following inequalities hold: h2 ν1 dx − g2+ ν1 dx < 0 and g2− ν1 dx − h2 ν1 dx < 0, (8) Ω
Ω
then problem (1) has at least one solution.
Ω
Ω
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Theorem 3. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold. If g1 , g2 satisfy (F1), (F2), (F3) and (6), problem (1) has at least one solution. Remark. Theorems 1 and 2 are the positive answers of an open problem in [1], and the conditions (7) and (8) are two technical assumptions to ensure the functional I defined in Section 2 is coercive in E1 . Theorem 3 is similar to the case p = q, which was discussed in [1]. 2. Proofs of theorems Define the functionals I, I1 , I2 : H → R as: β+1 λ1 (α + 1) α+1 |∇u|p dx + |∇v|q dx − a(x)|u|p dx I(u, v) = p q p Ω Ω Ω λ1 (β + 1) c(x)|v|q dx − λ1 b(x)|u|α |v|β uvdx − q Ω Ω − (α + 1)
G1 (x, u)dx − (β + 1) G2 (x, v)dx Ω + (α + 1) h1 udx + (β + 1) h2 vdx, Ω Ω (α + 1) p p α β I1 (u, v) = |∇u| dx − λ1 a(x)|u| dx − λ1 b(x)|u| |v| uvdx p Ω Ω Ω − (α + 1) G1 (x, u)dx + (α + 1) h1 udx, Ω Ω (β + 1) q q α β |∇v| dx − λ1 c(x)|v| dx − λ1 b(x)|u| |v| uvdx I2 (u, v) = q Ω Ω Ω − (β + 1) G2 (x, v)dx + (β + 1) h2 vdx, Ω
Ω
t
where G1 (x, t) = 0 g1 (x, s)ds, G2 (x, t) = from (2), it follows that
Ω
t 0
g2 (x, s)ds. From (F1), it is well known that I ∈ C 1 (W, R) and for all (u, v) ∈ H.
I(u, v) = I1 (u, v) + I2 (u, v)
From the variational view of point, a weak solution of problem (1) is exactly a critical point of the functional I in H. We will prove Theorem 1 by using the G-linking Theorem (see [5,9]) and the proof of Theorem 2 is similar. The proof of Theorem 3 is similar to the case p = q and the details are seen in [1]. Proof of Theorem 1. (i) The functional I satisfies the (P S) condition. Let (un , vn ) be a sequence such that there exists a positive constant c > 0 such that |I(un , vn )| ≤ c for all n,
and I ′ (un , vn ) → 0
as n → ∞.
(9)
We first say that (un , vn ) is bounded in H, and then prove that (un , vn ) has a convergent subsequence. Assume for contradiction that ∥(un , vn )∥ = ∥un ∥p + ∥vn ∥q → +∞ as n → ∞. Let Un := ∥un ∥pp + ∥vn ∥qq and n n , v¯n = v1/q , hence (¯ un , v¯n ) is bounded in H, i.e. ∥¯ un ∥pp + ∥¯ vn ∥qq = 1 for all n, which implies define u ¯n = u1/p Un
Un
there exists a subsequence of (¯ un , v¯n ), still denoted by (¯ un , v¯n ), and (¯ u, v¯) ∈ H such that (¯ un , v¯n ) ⇀ (¯ u, v¯) weakly in H, (¯ un , v¯n ) → (¯ u, v¯) strongly in Lp (Ω ) × Lq (Ω ) and (¯ un (x), v¯n (x)) → (¯ u(x), v¯(x)) for a.e. x ∈ Ω . Hence we can obtain |¯ un |p → |¯ u|p strongly in L1 (Ω ). it follows that p p a(x)|¯ u | dx − a(x)|¯ u | dx ≤ |a| ||¯ un |p − |¯ u|p |dx → 0 as n → ∞. (10) n ∞ Ω
Ω
Ω
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
Similarly we obtain q q c(x)|¯ vn | dx − c(x)|¯ v | dx ≤ |c|∞ ||¯ vn |q − |¯ v |q |dx → 0 Ω
Ω
α β α β b(x)|¯ →0 u | |¯ v | u ¯ v ¯ dx − b(x)|¯ u | |¯ v | u ¯ v ¯ dx n n n n Ω
(11)
as n → ∞.
(12)
Ω
From (3), (F1) and H¨ older’s inequality, we obtain G1 (x, u)dx ≤ |G (x, u)| dx = 1 Ω
as n → ∞,
Ω
Ω
Ω
|u|
0
for all v ∈ W01,q (Ω ),
Ω
1 Un
h|u|dx ≤ C1 ∥u∥p
G2 (x, vn )dx → 0
as n → ∞
(15)
Ω
From (9), we have n→∞
(14)
and from h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and H¨older’s inequality, it follows that 1 (h1 un + h2 vn )dx → 0 as n → ∞. Un Ω
lim sup
(13)
Ω
for all u ∈ W01,p (Ω ), similarly, we have G2 (x, v)dx ≤ C1 ∥v∥q where C1 is a positive constant. Hence we get 1 G1 (x, un )dx → 0 and Un Ω
|g1 (x, s)| dsdx ≤
I(un , vn ) ≤ 0, Un
hence from (10), (11), (12), (15), and (16), it follows that β+1 α+1 |∇¯ un |p dx + |∇¯ vn |q dx lim sup p q n→∞ Ω Ω α+1 β + 1 ≤ λ1 a(x)|¯ u|p dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx . p q Ω Ω Ω Hence, using the weak lower semicontinuity of the norm and the Poincar´ e inequality, we obtain α+1 β+1 p q α β a(x)|¯ u| dx + c(x)|¯ v | dx + b(x)|¯ u| |¯ v| u ¯v¯dx λ1 p q Ω Ω Ω α+1 β+1 ≤ |∇¯ u|p dx + |∇¯ v |q dx p q Ω Ω β+1 α+1 p ≤ lim inf |∇¯ un | dx + |∇¯ vn |q dx n→∞ p q Ω Ω α+1 β + 1 ≤ lim sup |∇¯ un |p dx + |∇¯ vn |q dx p q n→∞ Ω Ω α+1 β + 1 ≤ λ1 a(x)|¯ u|p dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx , p q Ω Ω Ω which implies that the following equality holds: α+1 β+1 |∇¯ u|p dx + |∇¯ v |q dx p q Ω Ω α+1 β+1 p = λ1 a(x)|¯ u| dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx , p q Ω Ω Ω
(16)
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
and by the uniform convexity of H, it follows that (¯ un , v¯n ) converges strongly to (¯ u, v¯) in H, and from the definition of (µ1 , ν1 ), it follows that (¯ u, v¯) = (±µ1 , ±ν1 ). In the following, assume that (¯ u, v¯) = (µ1 , ν1 ), and the case where (¯ u, v¯) = −(µ1 , ν1 ) may be treated similarly. Hence from the definitions of I1 , I2 , we have v¯n pI1 (un , vn ) qI2 (un , vn ) u ¯n ′ , + − I (un , vn ), 1/p 1/q p−1 q−1 (p − 1)Un (q − 1)Un α+1 p = g1 (x, un )¯ un dx − 1/p G1 (x, un )dx p−1 Ω Ω Un β+1 q + g2 (x, vn )¯ vn dx − 1/q G2 (x, vn )dx q−1 Ω Ω Un + (α + 1) h1 u ¯n dx + (β + 1) h1 v¯n dx. (17) Ω
From h1 ∈ L
p/(p−1)
Ω
q/(q−1)
(Ω ), h2 ∈ L
(Ω ), we observe h1 µ1 dx and h2 v¯n dx → h2 ν1 dx
h1 u ¯n dx →
Ω
Ω
Ω
From (F2) and (F3), we have g1 (x, un )¯ un dx → g1+ µ1 dx Ω
as n → ∞.
g2 (x, vn )¯ vn dx →
and
Ω
(18)
Ω
Ω
Ω
g2+ ν1 dx
as n → ∞. Finally, from Lebesgue dominated convergence theorem and (F2), we have G1 (x, un ) un 1 G (x, u )dx = dx → g1+ µ1 dx as n → ∞. 1 n 1/p 1/p u n Ω Ω Ω Un Un Similarly, we obtain 1 G (x, v )dx → g2+ ν1 dx as n → ∞. 2 n 1/q Ω Ω Un
(19)
(20)
(21)
Therefore, taking the limit in (17) and from (9), (18), (19), (20) and (21), we get (α + 1) h1 µ1 dx + (β + 1) h2 ν1 dx = (α + 1) g1+ µ1 dx + (β + 1) g2+ ν1 dx, Ω
Ω
Ω
Ω
which is a contradiction with the condition (5). Hence, (un , vn ) is bounded in H. From (F1), a standard argument implies that (un , vn ) strongly converges in H. (ii) The functional I satisfies the geometries of G-linking Theorem. From the definition of E1 = {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}, for any (u, v) ∈ E1 , we have α+1 β+1 |∇u|p dx + |∇v|q dx p q Ω Ω α+1 β+1 p q α β = λ1 a(x)|u| dx + c(x)|v| dx + b(x)|u| |v| uvdx . (22) p q Ω Ω Ω Moreover, we have 1 p G1 (x, t µ1 )dx = Ω
Ω
Ω
1
0
G2 (x, t1/q ν1 )dx = t1/q
1/p
g1 (x, τ t
µ1 )(t
1/p
1/p
µ1 )dτ dx = t
Ω
Ω
0
1
g2 (x, τ t1/q ν1 )ν1 dτ dx.
0
1
g1 (x, τ t1/p µ1 )µ1 dτ dx,
(23) (24)
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
From Lebesgue dominated convergence theorem, (F1) and (F2), we obtain 1
lim
t→+∞
1/p
g1 (x, τ t
0
Ω
µ1 )µ1 dτ dx = Ω
g1+ µ1 dx.
(25)
g2+ ν1 dx.
(26)
Similarly, lim
t→+∞
Ω
1
1/q
g2 (x, τ t
0
ν1 )ν1 dτ dx = Ω
Hence, from the definition of I, (22)–(24), it follows that I(t1/p µ1 , t1/q ν1 ) = (α + 1) (t1/p h1 µ1 − G1 (x, t1/p µ1 ))dx + (β + 1) (t1/q h2 ν1 − G2 (x, t1/q ν1 ))dx Ω Ω 1 = t1/p (α + 1) h1 µ1 − g1 (x, τ t1/p µ1 )µ1 dτ dx 0
Ω
+t
1/q
h2 ν1 −
(β + 1) Ω
0
1
1/q
g2 (x, τ t
ν1 )ν1 dτ
dx.
From the above equality, 1 < p < q, (7), (25) and (26), we have I(t1/p µ1 , t1/q ν1 ) → −∞ as t → +∞. Similarly, if g1+ and g2+ exchanged with g1− and g2− respectively, we obtain I(−t1/p µ1 , −t1/q ν1 ) → −∞ as t → +∞. Hence, we have lim I(±t1/p µ1 , ±t1/q ν1 ) = −∞.
(27)
t→+∞
On the other hand, let Λ2 := {(u, v) ∈ H : φ(u, v) ≥ λ2 ϕ(u, v)}. From (3), (13), (14), and H¨older’s inequality, for all (u, v) ∈ Λ2 , we obtain λ2 − λ1 β + 1 λ2 − λ1 α + 1 p |u| + |h | q |v| ∥u∥pp + ∥v∥qq − C1 (∥u∥p + ∥v∥q ) − |h1 | p−1 I(u, v) ≥ p 2 q−1 q λ2 p λ2 q λ2 − λ1 α+1 β+1 ≥ min , (∥u∥pp + ∥v∥qq ) − C2 (∥u∥p + ∥v∥q ), λ2 p q p , |h | q }, combining the above expression with (27), there is a constant where C2 = C1 + C max{|h1 | p−1 2 q−1 T > 0 such that
α0 := sup I(±t1/p µ1 , ±t1/q ν1 ) < β0 := t≥T
inf
(u,v)∈Λ2
I(u, v).
(28)
Let M = {(±t1/p µ1 , ±t1/q ν1 ) : t ≥ T } and G = {h ∈ C(S 0 , H) : h is odd and h(S 0 ) ⊂ M }, where S 0 is the boundary of the closed unit ball B 1 in R1 , that is S 0 = ∂B 1 . For any h ∈ G, by (28), we get h(S 0 ) ∩ Λ2 = Ø, which shows that G is a subset of C(S 0 , H \ Λ2 ). Let Γ = {h ∈ C(B 1 , H) : h|S 0 ∈ G}, we can claim: Γ is nonempty, and Λ2 and S 0 are G-linking, that is h(B 1 ) ∩ Λ2 ̸= Ø for all h ∈ Γ . The similar proof of conclusion may be found in [2,5,9], therefore we omit it. Finally, Theorem 1 is proved from the G-linking Theorem.
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Z.-Q. Ou / Applied Mathematics Letters 50 (2015) 29–36
References [1] N.B. Zographopoulos, p-Laplacian systems on resonance, Appl. Anal. 83 (5) (2004) 509–519. [2] Z.Q. Ou, C.L. Tang, Resonance problems for the p-Laplacian systems, J. Math. Anal. Appl. 345 (1) (2008) 511–521. [3] M. Landesman, A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970) 609–623. [4] D. Arcoya, L. Orsina, Landesman–Lazer conditions and quasilinear elliptic equations, Nonliear Anal. TMA 28 (10) (1997) 1623–1632. [5] P. Drabek, S.B. Robinson, Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1) (1999) 189–200. [6] C.L. Tang, Solvability for two-point boundary value problems, J. Math. Anal. Appl. 216 (1) (1997) 368–374. [7] C.L. Tang, Solvability of the forced Duffing equation at resonance, J. Math. Anal. Appl. 219 (1) (1998) 110–124. [8] C.L. Tang, Solvability of Neumann problem for elliptic equations at resonance, Nonlinear Anal. TMA 44 (3) (2001) 323–335. [9] S.Z. Song, C.L. Tang, Resonance problems for the p-Laplacian with a nonlinear boundary condition, Nonlinear Anal. TMA 64 (9) (2006) 2007–2021. [10] E.D. Da Silva, Multiplicity of solutions for gradient systems using Landesman–Lazer conditions, Abstr. Appl. Anal. (2010) Article ID:237826.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue of Jacobi matrixsystems Existence of weak solutions for a problem class of (p, q)-Laplacian on resonance✩ with mixed data Zeng-Qi Ou
Ying Wei 1
School of Mathematics Department and Statistics, Southwest University, People’sand Republic of China of Mathematics, Nanjing Chongqing University 400715, of Aeronautics Astronautics, Nanjing 210016, PR China
article
info a r t i c l e
abstract i n f o
a b s t r a c t
Article history: The existences of weak solutions for a class of (p, q)-Laplacian systems on resonance Received 21 April 2015 Article history: In certain this paper, the inverse eigenvalueconditions problem of are obtained under the Landesman–Lazer-type byreconstructing using G-link 16 January 2014 Received in revised formReceived 1 June 2015 a Jacobi matrix (1999). from its eigenvalues, leading answer principal Theorem due to Dr´ abek and Robinson Our results giveits a positive to Accepted 20 September 2014 Accepted 2 June 2015 submatrix and(2004). part of the eigenvalues of its submatrix an open problem of Zographopoulos online 22 October 2014 Available online 10 JuneAvailable 2015 is considered. The necessary and sufficient conditions for © 2015 Elsevier Ltd. All rights reserved. Submitted by Y. Wei
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
Keywords: MSC: Elliptic systems 15A18 The Landesman–Lazer-type 15A57 conditions Critical point Keywords: G-linking Theorem Jacobi matrix Eigenvalue Inverse problem Submatrix
1. Introduction and main results Consider the existence of weak solutions for the following elliptic systems p−2 α β −∆p u = λ1 a(x)|u| u + λ1 b(x)|u| |v| v + g1 (x, u) − h1 (x) −∆q v = λ1 c(x)|v|q−2 v + λ1 b(x)|u|α |v|β u + g2 (x, v) − h2 (x) u = v = 0
in Ω , in Ω , on ∂Ω ,
(1)
where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary, ∆p u = div(|∇u|p−2 ∇u) is the pLaplacian operator. 1 < p, q < N and α ≥ 0, β ≥ 0 satisfying α+1
+
β+1
= 1.
p q E-mail address: [email protected]. 1 Tel.: +86 13914485239. The nonlinearities g1 , g2 ∈ C(Ω × R, R) and h1 ∈ Lp/(p−1) (Ω ), h2 a, b, c ∈ C(Ω ) ∩ L∞http://dx.doi.org/10.1016/j.laa.2014.09.031 (Ω ) satisfy one of the following conditions: 0024-3795/© 2014 Published by Elsevier Inc.
(2) ∈ Lq/(q−1) (Ω ). The coefficient functions
✩ Supported by National Natural Science Foundation of China (No. 11471267) and the Fundamental Research Funds for the Central Universities (No. XDJK2011C039), the Postdoctoral Research Foundation of Chongqing (No. Xm201319). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.aml.2015.06.004 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
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(a1) a+ ̸= 0, where a+ (x) := max{a(x), 0}; (a2) c+ = ̸ 0; (a3) a = c = 0 and b+ ̸= 0. 1 Let W01,p (Ω ) be the Sobolev space with norm ∥u∥p = Ω |∇u|p dx p for any u ∈ W01,p (Ω ). From the p , the embedding W01,p (Ω ) ↩→ Lτ (Ω ) is Sobolev and Rellich embedding theorems, for any 1 ≤ τ < NN−p continuous and compact, and there is a positive constant C = C(τ, p, N ) such that |u|τ ≤ C∥u∥p ,
for all u ∈ W01,p (Ω ),
(3)
where | · |τ denotes the norm of Lτ (Ω ). In the sequel, C also denotes a positive constant related to the above embedding. Let H = W01,p (Ω ) × W01,q (Ω ) be Banach space with the norm ∥(u, v)∥ = ∥u∥p + ∥v∥q for any (u, v) ∈ H. Let us consider the following nonlinear eigenvalue problem with weights p−2 α β −∆p u = λa(x)|u| u + λb(x)|u| |v| v in Ω , q−2 α β (4) −∆q v = λc(x)|v| v + λb(x)|u| |v| u in Ω , u = v = 0 on ∂Ω . Define the functionals φ, ϕ : H → R as α+1 β+1 φ(u, v) = |∇u|p dx + |∇v|q dx, p q Ω Ω β+1 α+1 p a(x)|u| dx + c(x)|v|q dx + b(x)|u|α |v|β uvdx, ϕ(u, v) = p q Ω Ω Ω and the manifold Σ = {(u, v) ∈ H : ψ(u, v) = 1}. We say that φ(u, v), ϕ(u, v) are (p, q)-homogeneous, namely φ(t1/p u, t1/q v) = tφ(u, v),
ϕ(t1/p u, t1/q v) = tϕ(u, v)
for any t > 0 and (u, v) ∈ H,
and if one of (a1)–(a3) is satisfied, Σ is a symmetric nonempty manifold in H. By a standard argument (the similar proof seen in [1,2]), problem (4) has a sequence of eigenvalues with the variational characterization λk = inf
sup φ(u, v),
F ∈Σk (u,v)∈F
where Σk = {F ⊂ Σ : there is an odd, continuous and surjective γ : S k−1 → F } and S k−1 denotes the unit sphere in Rk . On the other hand, let α+1 β+1 λ′1 = inf |∇u|p dx + |∇v|q dx, p q (u,v)∈Σ Ω Ω then λ′1 is also the first eigenvalue of (4), and from the definition of λ1 , it is easy to see that λ1 = λ′1 . Moreover, λ1 is a simple, isolated and positive principal eigenvalue of (4) and corresponds to a positive normalized eigenvalue (µ0 , ν0 ), that is ∥µ0 ∥p + ∥ν0 ∥q = 1, hence the set of all eigenfunctions corresponding to λ1 is E1 := {(t1/p µ0 , t1/q ν0 ) : t ≥ 0} ∪ {(−t1/p µ0 , −t1/q ν0 ) : t ≥ 0}. Since φ(u, v), ϕ(u, v) are (p, q)-homogeneous, there exists a positive constant t0 such that 1/p
1/q
∥t0 µ0 ∥pp + ∥t0 ν0 ∥qq = 1. 1/p
1/q
Let µ1 = t0 µ0 , ν1 = t0 ν0 , then {(t1/p µ0 , t1/q ν0 ) : t ≥ 0}∪{(−t1/p µ0 , −t1/q ν0 ) : t ≥ 0} = {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}, In the following, let E1 := {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}.
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The set E1 is not a one-dimensional linear subspace of H and the corresponding orthogonal decomposition on H does not hold with respect to the first eigenvalue λ1 . The Landesman–Lazer-type conditions were introduced by Landesman and Lazer in [3] and have been widely used and extended (see [1,2,4–10] and the reference therein). In [1], if p = q ≥ 2 and the certain Landesman–Lazer-type conditions hold, Zographopoulos proved that problem (1) has at least one weak solution, but he does not know whether the similar result holds for p ̸= q. In this paper, we will give out a positive answer. We first state some auxiliary conditions. (F1) There is h ∈ C(Ω , R+ ) such that |g1 (x, t)| ≤ h(x)
and |g2 (x, t)| ≤ h(x) ∀ (x, t) ∈ Ω × R;
(F2) There exist functions g1+ , g1− ∈ C(Ω , R) such that g1+ (x) = lim g1 (x, t),
g1− (x) = lim g1 (x, t)
t→+∞
t→−∞
uniformly for all x ∈ Ω ;
(F3) There exist functions g2+ , g2− ∈ C(Ω , R) such that g2+ (x) = lim g2 (x, t),
g2− (x) = lim g2 (x, t)
t→+∞
t→−∞
uniformly for all x ∈ Ω .
The Landesman–Lazer-type conditions for problem (1) are assumed either (α + 1) g1− µ1 dx + (β + 1) g2− ν1 dx < (α + 1) h1 µ1 dx + (β + 1) h2 ν1 dx Ω
Ω
Ω
< (α + 1) Ω
Ω
g1+ µ1 dx + (β + 1)
Ω
g2+ ν1 dx;
(5)
or (α + 1) Ω
g1− µ1 dx + (β + 1)
Ω
g2− ν1 dx > (α + 1)
h1 µ1 dx + (β + 1)
Ω
> (α + 1) Ω
h2 ν1 dx Ω
g1+ µ1 dx + (β + 1)
Ω
g2+ ν1 dx.
(6)
The main results of this paper are the following theorems. Theorem 1. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold, and g1 , g2 satisfy (F1), (F2), (F3) and (5). If 1 < p < q, assume that the following inequalities hold: − + h1 µ1 dx − g1 µ1 dx < 0 and g1 µ1 dx − h1 µ1 dx < 0, (7) Ω
Ω
Ω
Ω
then problem (1) has at least one solution. In the other case 1 < q < p, we have the following result. Theorem 2. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold, and g1 , g2 satisfy (F1), (F2), (F3) and (5). If 1 < q < p, assume that the following inequalities hold: h2 ν1 dx − g2+ ν1 dx < 0 and g2− ν1 dx − h2 ν1 dx < 0, (8) Ω
Ω
then problem (1) has at least one solution.
Ω
Ω
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Theorem 3. Suppose that (2), h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and one of (a1)–(a3) hold. If g1 , g2 satisfy (F1), (F2), (F3) and (6), problem (1) has at least one solution. Remark. Theorems 1 and 2 are the positive answers of an open problem in [1], and the conditions (7) and (8) are two technical assumptions to ensure the functional I defined in Section 2 is coercive in E1 . Theorem 3 is similar to the case p = q, which was discussed in [1]. 2. Proofs of theorems Define the functionals I, I1 , I2 : H → R as: β+1 λ1 (α + 1) α+1 |∇u|p dx + |∇v|q dx − a(x)|u|p dx I(u, v) = p q p Ω Ω Ω λ1 (β + 1) c(x)|v|q dx − λ1 b(x)|u|α |v|β uvdx − q Ω Ω − (α + 1)
G1 (x, u)dx − (β + 1) G2 (x, v)dx Ω + (α + 1) h1 udx + (β + 1) h2 vdx, Ω Ω (α + 1) p p α β I1 (u, v) = |∇u| dx − λ1 a(x)|u| dx − λ1 b(x)|u| |v| uvdx p Ω Ω Ω − (α + 1) G1 (x, u)dx + (α + 1) h1 udx, Ω Ω (β + 1) q q α β |∇v| dx − λ1 c(x)|v| dx − λ1 b(x)|u| |v| uvdx I2 (u, v) = q Ω Ω Ω − (β + 1) G2 (x, v)dx + (β + 1) h2 vdx, Ω
Ω
t
where G1 (x, t) = 0 g1 (x, s)ds, G2 (x, t) = from (2), it follows that
Ω
t 0
g2 (x, s)ds. From (F1), it is well known that I ∈ C 1 (W, R) and for all (u, v) ∈ H.
I(u, v) = I1 (u, v) + I2 (u, v)
From the variational view of point, a weak solution of problem (1) is exactly a critical point of the functional I in H. We will prove Theorem 1 by using the G-linking Theorem (see [5,9]) and the proof of Theorem 2 is similar. The proof of Theorem 3 is similar to the case p = q and the details are seen in [1]. Proof of Theorem 1. (i) The functional I satisfies the (P S) condition. Let (un , vn ) be a sequence such that there exists a positive constant c > 0 such that |I(un , vn )| ≤ c for all n,
and I ′ (un , vn ) → 0
as n → ∞.
(9)
We first say that (un , vn ) is bounded in H, and then prove that (un , vn ) has a convergent subsequence. Assume for contradiction that ∥(un , vn )∥ = ∥un ∥p + ∥vn ∥q → +∞ as n → ∞. Let Un := ∥un ∥pp + ∥vn ∥qq and n n , v¯n = v1/q , hence (¯ un , v¯n ) is bounded in H, i.e. ∥¯ un ∥pp + ∥¯ vn ∥qq = 1 for all n, which implies define u ¯n = u1/p Un
Un
there exists a subsequence of (¯ un , v¯n ), still denoted by (¯ un , v¯n ), and (¯ u, v¯) ∈ H such that (¯ un , v¯n ) ⇀ (¯ u, v¯) weakly in H, (¯ un , v¯n ) → (¯ u, v¯) strongly in Lp (Ω ) × Lq (Ω ) and (¯ un (x), v¯n (x)) → (¯ u(x), v¯(x)) for a.e. x ∈ Ω . Hence we can obtain |¯ un |p → |¯ u|p strongly in L1 (Ω ). it follows that p p a(x)|¯ u | dx − a(x)|¯ u | dx ≤ |a| ||¯ un |p − |¯ u|p |dx → 0 as n → ∞. (10) n ∞ Ω
Ω
Ω
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Similarly we obtain q q c(x)|¯ vn | dx − c(x)|¯ v | dx ≤ |c|∞ ||¯ vn |q − |¯ v |q |dx → 0 Ω
Ω
α β α β b(x)|¯ →0 u | |¯ v | u ¯ v ¯ dx − b(x)|¯ u | |¯ v | u ¯ v ¯ dx n n n n Ω
(11)
as n → ∞.
(12)
Ω
From (3), (F1) and H¨ older’s inequality, we obtain G1 (x, u)dx ≤ |G (x, u)| dx = 1 Ω
as n → ∞,
Ω
Ω
Ω
|u|
0
for all v ∈ W01,q (Ω ),
Ω
1 Un
h|u|dx ≤ C1 ∥u∥p
G2 (x, vn )dx → 0
as n → ∞
(15)
Ω
From (9), we have n→∞
(14)
and from h1 ∈ Lp/(p−1) (Ω ), h2 ∈ Lq/(q−1) (Ω ) and H¨older’s inequality, it follows that 1 (h1 un + h2 vn )dx → 0 as n → ∞. Un Ω
lim sup
(13)
Ω
for all u ∈ W01,p (Ω ), similarly, we have G2 (x, v)dx ≤ C1 ∥v∥q where C1 is a positive constant. Hence we get 1 G1 (x, un )dx → 0 and Un Ω
|g1 (x, s)| dsdx ≤
I(un , vn ) ≤ 0, Un
hence from (10), (11), (12), (15), and (16), it follows that β+1 α+1 |∇¯ un |p dx + |∇¯ vn |q dx lim sup p q n→∞ Ω Ω α+1 β + 1 ≤ λ1 a(x)|¯ u|p dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx . p q Ω Ω Ω Hence, using the weak lower semicontinuity of the norm and the Poincar´ e inequality, we obtain α+1 β+1 p q α β a(x)|¯ u| dx + c(x)|¯ v | dx + b(x)|¯ u| |¯ v| u ¯v¯dx λ1 p q Ω Ω Ω α+1 β+1 ≤ |∇¯ u|p dx + |∇¯ v |q dx p q Ω Ω β+1 α+1 p ≤ lim inf |∇¯ un | dx + |∇¯ vn |q dx n→∞ p q Ω Ω α+1 β + 1 ≤ lim sup |∇¯ un |p dx + |∇¯ vn |q dx p q n→∞ Ω Ω α+1 β + 1 ≤ λ1 a(x)|¯ u|p dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx , p q Ω Ω Ω which implies that the following equality holds: α+1 β+1 |∇¯ u|p dx + |∇¯ v |q dx p q Ω Ω α+1 β+1 p = λ1 a(x)|¯ u| dx + c(x)|¯ v |q dx + b(x)|¯ u|α |¯ v |β u ¯v¯dx , p q Ω Ω Ω
(16)
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and by the uniform convexity of H, it follows that (¯ un , v¯n ) converges strongly to (¯ u, v¯) in H, and from the definition of (µ1 , ν1 ), it follows that (¯ u, v¯) = (±µ1 , ±ν1 ). In the following, assume that (¯ u, v¯) = (µ1 , ν1 ), and the case where (¯ u, v¯) = −(µ1 , ν1 ) may be treated similarly. Hence from the definitions of I1 , I2 , we have v¯n pI1 (un , vn ) qI2 (un , vn ) u ¯n ′ , + − I (un , vn ), 1/p 1/q p−1 q−1 (p − 1)Un (q − 1)Un α+1 p = g1 (x, un )¯ un dx − 1/p G1 (x, un )dx p−1 Ω Ω Un β+1 q + g2 (x, vn )¯ vn dx − 1/q G2 (x, vn )dx q−1 Ω Ω Un + (α + 1) h1 u ¯n dx + (β + 1) h1 v¯n dx. (17) Ω
From h1 ∈ L
p/(p−1)
Ω
q/(q−1)
(Ω ), h2 ∈ L
(Ω ), we observe h1 µ1 dx and h2 v¯n dx → h2 ν1 dx
h1 u ¯n dx →
Ω
Ω
Ω
From (F2) and (F3), we have g1 (x, un )¯ un dx → g1+ µ1 dx Ω
as n → ∞.
g2 (x, vn )¯ vn dx →
and
Ω
(18)
Ω
Ω
Ω
g2+ ν1 dx
as n → ∞. Finally, from Lebesgue dominated convergence theorem and (F2), we have G1 (x, un ) un 1 G (x, u )dx = dx → g1+ µ1 dx as n → ∞. 1 n 1/p 1/p u n Ω Ω Ω Un Un Similarly, we obtain 1 G (x, v )dx → g2+ ν1 dx as n → ∞. 2 n 1/q Ω Ω Un
(19)
(20)
(21)
Therefore, taking the limit in (17) and from (9), (18), (19), (20) and (21), we get (α + 1) h1 µ1 dx + (β + 1) h2 ν1 dx = (α + 1) g1+ µ1 dx + (β + 1) g2+ ν1 dx, Ω
Ω
Ω
Ω
which is a contradiction with the condition (5). Hence, (un , vn ) is bounded in H. From (F1), a standard argument implies that (un , vn ) strongly converges in H. (ii) The functional I satisfies the geometries of G-linking Theorem. From the definition of E1 = {(t1/p µ1 , t1/q ν1 ) : t ≥ 0} ∪ {(−t1/p µ1 , −t1/q ν1 ) : t ≥ 0}, for any (u, v) ∈ E1 , we have α+1 β+1 |∇u|p dx + |∇v|q dx p q Ω Ω α+1 β+1 p q α β = λ1 a(x)|u| dx + c(x)|v| dx + b(x)|u| |v| uvdx . (22) p q Ω Ω Ω Moreover, we have 1 p G1 (x, t µ1 )dx = Ω
Ω
Ω
1
0
G2 (x, t1/q ν1 )dx = t1/q
1/p
g1 (x, τ t
µ1 )(t
1/p
1/p
µ1 )dτ dx = t
Ω
Ω
0
1
g2 (x, τ t1/q ν1 )ν1 dτ dx.
0
1
g1 (x, τ t1/p µ1 )µ1 dτ dx,
(23) (24)
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From Lebesgue dominated convergence theorem, (F1) and (F2), we obtain 1
lim
t→+∞
1/p
g1 (x, τ t
0
Ω
µ1 )µ1 dτ dx = Ω
g1+ µ1 dx.
(25)
g2+ ν1 dx.
(26)
Similarly, lim
t→+∞
Ω
1
1/q
g2 (x, τ t
0
ν1 )ν1 dτ dx = Ω
Hence, from the definition of I, (22)–(24), it follows that I(t1/p µ1 , t1/q ν1 ) = (α + 1) (t1/p h1 µ1 − G1 (x, t1/p µ1 ))dx + (β + 1) (t1/q h2 ν1 − G2 (x, t1/q ν1 ))dx Ω Ω 1 = t1/p (α + 1) h1 µ1 − g1 (x, τ t1/p µ1 )µ1 dτ dx 0
Ω
+t
1/q
h2 ν1 −
(β + 1) Ω
0
1
1/q
g2 (x, τ t
ν1 )ν1 dτ
dx.
From the above equality, 1 < p < q, (7), (25) and (26), we have I(t1/p µ1 , t1/q ν1 ) → −∞ as t → +∞. Similarly, if g1+ and g2+ exchanged with g1− and g2− respectively, we obtain I(−t1/p µ1 , −t1/q ν1 ) → −∞ as t → +∞. Hence, we have lim I(±t1/p µ1 , ±t1/q ν1 ) = −∞.
(27)
t→+∞
On the other hand, let Λ2 := {(u, v) ∈ H : φ(u, v) ≥ λ2 ϕ(u, v)}. From (3), (13), (14), and H¨older’s inequality, for all (u, v) ∈ Λ2 , we obtain λ2 − λ1 β + 1 λ2 − λ1 α + 1 p |u| + |h | q |v| ∥u∥pp + ∥v∥qq − C1 (∥u∥p + ∥v∥q ) − |h1 | p−1 I(u, v) ≥ p 2 q−1 q λ2 p λ2 q λ2 − λ1 α+1 β+1 ≥ min , (∥u∥pp + ∥v∥qq ) − C2 (∥u∥p + ∥v∥q ), λ2 p q p , |h | q }, combining the above expression with (27), there is a constant where C2 = C1 + C max{|h1 | p−1 2 q−1 T > 0 such that
α0 := sup I(±t1/p µ1 , ±t1/q ν1 ) < β0 := t≥T
inf
(u,v)∈Λ2
I(u, v).
(28)
Let M = {(±t1/p µ1 , ±t1/q ν1 ) : t ≥ T } and G = {h ∈ C(S 0 , H) : h is odd and h(S 0 ) ⊂ M }, where S 0 is the boundary of the closed unit ball B 1 in R1 , that is S 0 = ∂B 1 . For any h ∈ G, by (28), we get h(S 0 ) ∩ Λ2 = Ø, which shows that G is a subset of C(S 0 , H \ Λ2 ). Let Γ = {h ∈ C(B 1 , H) : h|S 0 ∈ G}, we can claim: Γ is nonempty, and Λ2 and S 0 are G-linking, that is h(B 1 ) ∩ Λ2 ̸= Ø for all h ∈ Γ . The similar proof of conclusion may be found in [2,5,9], therefore we omit it. Finally, Theorem 1 is proved from the G-linking Theorem.
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