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Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces
Linear Algebra its Applications 466315–329 (2015) 102–116 Nonlinear Analysis: Real Worldand Applications 26 (2015)
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Linear Algebra and its Applications Nonlinear Analysis: Real World Applications www.elsevier.com/locate/laa www.elsevier.com/locate/nonrwa
eigenvalue Solutions for aInverse quasilinear ellipticproblem equationofinJacobi matrix withspaces mixed data Musielak–Sobolev YingZhao Wei 1 Duchao Liu ∗ , Peihao School of Mathematics Department and Statistics, Lanzhou University, 730000, PR China and Astronautics, of Mathematics, NanjingLanzhou University of Aeronautics Nanjing 210016, PR China
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abstract i n f o
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Article history: Critical point theory is used to show the existence of weak solutions to a quasilinear Article history: Received 16 August 2014 In thisunder paper, the inverse eigenvalue of reconstructing elliptic differential equation the functional frameworkproblem of the Musielak–Sobolev Received 16 January 2014 Received in revised form 12 June a Jacobi matrix from its eigenvalues, its leading principal spaces in a bounded smooth domain with Dirichlet boundary condition. Accepted 20 September 2014 2015 submatrix and part© of theElsevier eigenvalues of rights its submatrix 2015 Ltd. All reserved. Accepted 23 June 2015 Available online 22 October 2014 is considered. The necessary and sufficient conditions for Submitted by Y. Wei Dedicated to Professor Xianling Fan MSC: on the occasion of his 70th birthday 15A18 15A57 Keywords: Musielak–Sobolev space Keywords: Critical point theory Jacobi matrix Mountain Pass type solution Eigenvalue Fountain theorem Inverse problem Submatrix
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
1. Introduction In this paper we deal with the existence of weak solutions to a differential equation in the Musielak–Sobolev spaces of the form −div b(x, |∇u|)∇u = f (x, u) in Ω , (1.1) u=0 on ∂Ω , where Ω ⊂ RN is a bounded smooth domain. In the study of nonlinear differential equations, it is well known that more general functional space can handle differential equations with more nonlinearities. If we want to study a general form of differential E-mail address: [email protected]. 1 Tel.: +86 13914485239. equations, it is very important to find a proper functional space corresponding to their solutions. Differential equations in variable exponent Sobolev spaces and Orlicz–Sobolev spaces have been http://dx.doi.org/10.1016/j.laa.2014.09.031 Published by Elsevier Inc.best knowledge, however, differential equations in studied extensively 0024-3795/© in recent 2014 years, see [1–4]. To our ∗ Corresponding author. Tel.: +86 13893289235; fax: +86 09318912481. E-mail addresses: [email protected] (D. Liu), [email protected] (P. Zhao).
http://dx.doi.org/10.1016/j.nonrwa.2015.06.002 1468-1218/© 2015 Elsevier Ltd. All rights reserved.
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Musielak–Sobolev spaces have been studied little. In the research of the paper [5], Benkirane and Sidi give an existence result for nonlinear elliptic equations in Musielak–Sobolev spaces, and in this paper the authors give an embedding theorem in Musielak–Sobolev spaces. In two recent papers (see [6,7]), Fan gives some properties about this kind of functional space. It is possible to make an application of those properties to give some existence of solutions to the elliptic equations with much more nonlinearities. Our aim in this paper is to study the existence of solutions to a kind of elliptic differential equation under the functional frame work of Musielak–Sobolev spaces, which is a more general case of the variable exponent Sobolev spaces and Orlicz–Sobolev spaces. The method we used here is the variational method. The paper is organized as follows. In Section 2, for the readers convenience we recall some definitions and properties about Musielak–Orlicz–Sobolev spaces. In Section 3, we give some basic properties of the differential operator in Musielak–Sobolev spaces corresponding to the equation. In Section 4, we give several existence results for the weak solutions to our problem (1.1). 2. The Musielak–Orlicz–Sobolev spaces In this section, we list some definitions and propositions (some of them are new in our case, e.g. Proposition 2.3(4), Remark 2.2, Remark 2.4 for Theorem 2.10) about Musielak–Orlicz–Sobolev spaces for the readers to refer. Firstly, we give the definition of “N -function” and “generalized N -function” as follows. Definition 2.1. A function A : R → [0, +∞) is called an N -function, denoted by A ∈ N , if A is even and convex, A(0) = 0, 0 < A(t) ∈ C 0 for t ̸= 0, and the following conditions hold lim
t→0+
A(t) =0 t
and
lim
t→+∞
A(t) = +∞. t
A function A : Ω × R → [0, +∞) is called a generalized N -function, denoted by A ∈ N (Ω ), if for each t ∈ [0, +∞), the function A(·, t) is measurable, and for a.e. x ∈ Ω , we have A(x, ·) ∈ N . Let A ∈ N (Ω ), the Musielak–Orlicz space LA (Ω ) is defined by |u(x)| LA (Ω ) := u : u is a measurable real function, and ∃λ > 0 such that A x, dx < +∞ λ Ω with the (Luxemburg) norm
∥u∥LA (Ω)
|u(x)| = ∥u∥A := inf λ > 0 : A x, λ Ω
dx ≤ 1 .
The Musielak–Sobolev space W 1,A (Ω ) can be defined by W 1,A (Ω ) := {u ∈ LA (Ω ) : |∇u| ∈ LA (Ω )} with the norm ∥u∥W 1,A (Ω) = ∥u∥1,A := ∥u∥A + ∥∇u∥A , where ∥∇u∥A := ∥ |∇u| ∥A . Definition 2.2. We say that A satisfies Condition (∆2 ), if there exist a positive constant K > 0 and a nonnegative function h ∈ L1 (Ω ) such that A(x, 2t) ≤ KA(x, t) + h(x)
for x ∈ Ω and t ∈ R.
A is called locally integrable if A(·, t0 ) ∈ L1 (Ω ) for every t0 > 0. For x ∈ Ω and t ≥ 0, we denote by a(x, t) the right-hand derivative of A(x, ·) at t, at the same time define |t| a(x, t) = −a(x, −t). Then A(x, t) = 0 a(x, s) ds for x ∈ Ω and t ∈ R.
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Define A′ : Ω × R → [0, +∞) by A′ (x, s) = sup st − A(x, t)
for x ∈ Ω and s ∈ R.
t∈R
A′ is called the complementary function to A in the sense of Young. It is well known that A′ ∈ N (Ω ) and A is also the complementary function to A′ . ′ For x ∈ Ω and s ≥ 0, we denote by a−1 + (x, s) the right-hand derivative of A (x, ·) at s at the same time −1 −1 define a+ (x, s) = −a+ (x, −s) for x ∈ Ω and s ≤ 0. Then for x ∈ Ω and s ≥ 0, we have a−1 + (x, s) = sup{t ≥ 0 : a(x, t) ≤ s} = inf{t > 0 : a(x, t) > s}. Proposition 2.1. Let A ∈ N (Ω ). Then the following assertions hold. (1) A(x, t) ≤ a(x, t)t ≤ A(x, 2t) for x ∈ Ω and t ∈ R; (2) A and A′ satisfy the Young inequality st ≤ A(x, t) + A′ (x, s) for x ∈ Ω and s, t ∈ R and the equality holds if s = a(x, t) or t = a−1 + (x, s). Let A, B ∈ N (Ω ). We say that A is weaker than B, denoted by A 4 B, if there exist positive constants K1 , K2 and a nonnegative function h ∈ L1 (Ω ) such that A(x, t) ≤ K1 B(x, K2 t) + h(x)
for x ∈ Ω and t ∈ [0, +∞).
(2.1) ′
′
Proposition 2.2. Let A, B ∈ N (Ω ) and A 4 B. Then B ′ 4 A′ , LA (Ω ) ↩→ LB (Ω ) and LB (Ω ) ↩→ LA (Ω ). Proposition 2.3. Let A ∈ N (Ω ) satisfy (∆2 ). Then the following assertions hold, (1) (2) (3) (4) (5)
LA (Ω ) = {u : u is a measurable function, and Ω A(x, |u(x)|) dx < +∞}; A(x, |u|) dx < 1(resp. = 1; > 1) ⇐⇒ ∥u∥A < 1(resp. = 1; > 1), where u ∈ LA (Ω ); Ω A(x, |un |) dx → 0(resp. 1; +∞) ⇐⇒ ∥un ∥A → 0(resp. 1; +∞), where {un } ∈ LA (Ω ); Ω A un → u in L (Ω ) =⇒ Ω A(x, |un |) dx − A(x, |u|) dx → 0 as n → ∞; If A′ also satisfies (∆2 ), then u(x)v(x) dx ≤ 2∥u∥A ∥v∥A′ ,
′
∀u ∈ LA (Ω ), v ∈ LA (Ω );
Ω
′
(6) a(·, |u(·)|) ∈ LA (Ω ) for every u ∈ LA (Ω ). Proof. The readers can find the proofs of (1–3) and (5–6) in [6]. For the item (4), in the Ref. [6], professor Fan only proves that “(4′ ) un → u in LA (Ω ) =⇒ Ω A(x, |un |) dx → Ω A(x, |u|) dx as n → ∞”, we can prove that this conclusion is equivalent to the conclusion in Proposition 2.3(4). It is trivial that (4) =⇒ (4′ ) is right. We can prove (4′ ) =⇒ (4). In fact, if we set un , for x ∈ {|un | ≥ |u|}, + un = u, for x ∈ {|un | < |u|}; u, for x ∈ {|un | ≥ |u|}, u− n = un , for x ∈ {|un | < |u|},
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A then u± n → u in L (Ω ). By the conclusion in Ref. [6], we have A(x, |u+ n |) − A(x, |u|) dx → 0,
(2.2)
Ω
and
A(x, |u− n |) − A(x, |u|) dx → 0,
(2.3)
Ω
as n → +∞. By taking (2.2)–(2.3), we can get − A(x, |un |) − A(x, |u|) dx = A(x, |u+ n |) dx − A(x, |un |) dx → 0, Ω
as n → +∞.
Ω
Proposition 2.4. Let A ∈ N (Ω ) be locally integrable. Then (LA (Ω ), ∥ · ∥) is a separable Banach space. The following assumptions will be used. (A1 ) inf x∈Ω A(x, 1) = c1 > 0; (A2 ) For every t0 > 0, there exists c = c(t0 ) > 0 such that A(x, t) ≥ c and t
A′ (x, t) ≥ c, ∀t ≥ t0 , x ∈ Ω . t
Remark 2.1. Obviously, (A2 ) ⇒ (A1 ). If A(x, t) = A(t) is an N -function, then (A1 ) and (A2 ) hold automatically, and A is automatically locally integrable. Proposition 2.5. If A ∈ N (Ω ) satisfies (A1 ), then LA (Ω ) ↩→ L1 (Ω ) and W 1,A (Ω ) ↩→ W 1,1 (Ω ). Proposition 2.6 (See [6]). Let A ∈ N (Ω ), both A and A′ be locally integrable and satisfy ∆2 and (A2 ). Then ′ the space LA (Ω ) is reflexive, and the mapping J : LA (Ω ) → (LA (Ω ))∗ defined by ′ ⟨J(v), w⟩ = v(x)w(x) dx, ∀v ∈ LA (Ω ), w ∈ LA (Ω ) Ω
is a linear isomorphism and ∥J(v)∥(LA (Ω))∗ ≤ 2∥v∥LA′ (Ω) . Let A ∈ N (Ω ) be locally integrable. We will denote W01,A (Ω ) := C0∞ (Ω ) D01,A (Ω ) := C0∞ (Ω )
∥ · ∥W 1,A (Ω)
∥∇ · ∥LA (Ω)
.
In the case ∥∇u∥A is an equivalent norm in W01,A (Ω ), W01,A (Ω ) = D01,A (Ω ). Proposition 2.7 (See [6]). Let A ∈ N (Ω ) be locally integrable and satisfy (A1 ). Then (1) the spaces W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are separable Banach spaces, and W01,A (Ω ) ↩→ W 1,A (Ω ) ↩→ W 1,1 (Ω ) D01,A (Ω ) ↩→ D01,1 (Ω ) = W01,1 (Ω );
(2) the spaces W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are reflexive provided LA (Ω ) is reflexive.
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Proposition 2.8. Let A, B ∈ N (Ω ) and A be locally integrable. If there is a compact embedding W 1,A (Ω ) ↩→↩→ LB (Ω ) and A 4 B, then there holds the following Poincar´e inequality ∥u∥A ≤ c∥∇u∥A ,
∀u ∈ W01,A (Ω ),
which implies that ∥∇ · ∥A is an equivalent norm in W01,A (Ω ) and W01,A (Ω ) = D01,A (Ω ). The following assumptions will be used. (P1 ) Ω ⊂ RN (N ≥ 2) is a bounded domain with the cone property, and A ∈ N (Ω ); (P2 ) A : Ω × R → [0, +∞) is continuous and A(x, t) ∈ (0, +∞) for x ∈ Ω and t ∈ (0, +∞). Let A satisfy (P1 ) and (P2 ). Define A1 : Ω × [0, +∞) → [0, +∞) by A(x, 1)|t|, for x ∈ Ω and |t| ∈ [0, 1], A1 (x, t) = A(x, t), for x ∈ Ω and |t| > 1. Remark 2.2. Since Ω is bounded, we can see (see [8,7]) that LA (Ω ) = LA1 (Ω ) and W 1,A (Ω ) = W 1,A1 (Ω ). Sometimes, in the study of embeddings of W 1,A (Ω ), it is much more convenient to consider A1 instead of A. In other words, we may assume that A satisfies the following condition which is not essential because Ω is bounded. (P0 ) A(x, t) = A(x, 1)|t| for x ∈ Ω and |t| ∈ [0, 1). Under assumptions (P1 ) and (P2 ), for each x ∈ Ω we can define A∗ (x, t) for x ∈ Ω and t ≥ 0 as in [7], and define A∗ (x, t) = A∗ (x, −t) for x ∈ Ω and t ≤ 0. We call A∗ the Sobolev conjugate function of A. Set 1 lims→+∞ A−1 ∗ (x, s) = T (x), then A∗ (x, ·) ∈ C (0, T (x)). Furthermore for A ∈ N (Ω ) and T (x) = +∞ for any x ∈ Ω , it is known that A∗ ∈ N (Ω ) (see in [9]). Let X be a metric space and f : X → (−∞, +∞] be an extended real-valued function. For x ∈ X with f (x) ∈ R, the continuity of f at x is well defined. For x ∈ X with f (x) = +∞, we say that f is continuous at x if given any M > 0, there exists a neighborhood U of x such that f (y) > M for all y ∈ U . We say that f : X → (−∞, +∞] is continuous on X if f is continuous at every x ∈ X. Define Dom(f ) = {x ∈ X : f (x) ∈ R} and denote by C 1−0 (X) the set of all locally Lipschitz continuous real-valued functions defined on X. The following assumptions will also be used. (P3 ) T : Ω → [0, +∞] is continuous on Ω and T ∈ C 1−0 (Dom(T )); (P4 ) A∗ ∈ C 1−0 (Dom(A∗ )) and there exist positive constants δ0 < that |∇x A∗ (x, t)| ≤ C0 (A∗ (x, t))1+δ0 ,
1 N,
C0 and 0 < t0 < minx∈Ω T (x) such
j = 1, . . . , N,
for x ∈ Ω and |t| ∈ [t0 , T (x)) provided ∇x A∗ (x, t) exists. Let A, B ∈ N (Ω ). We say that A ≪ B if, for any k > 0, lim
t→+∞
A(x, kt) =0 B(x, t)
uniformly for x ∈ Ω .
Remark 2.3. Suppose that A, B ∈ N (Ω ), then A ≪ B ⇒ A 4 B. We give two embedding theorems for Musielak–Sobolev spaces developed by Fan in [7] with some suitable changes (see Remark 2.4) in order to be more convenient to make an application.
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Theorem 2.9. Let (P1 )–(P4 ) hold. Then (i) There is a continuous embedding W 1,A (Ω ) ↩→ LA∗ (Ω ); (ii) Suppose that B ∈ N (Ω ), B : Ω × [0, +∞) → [0, +∞) is continuous, and B(x, t) ∈ (0, +∞) for x ∈ Ω and t ∈ (0, +∞). If B ≪ A∗ , then there is a compact embedding W 1,A (Ω ) ↩→↩→ LB (Ω ). By Theorem 2.9, Remark 2.3 and Proposition 2.8, we have the following. Theorem 2.10. Let (P1 )–(P4 ) hold and furthermore, A, A∗ ∈ N (Ω ). Then (i) A ≪ A∗ , and there is a compact embedding W 1,A (Ω ) ↩→↩→ LA (Ω ); (ii) there holds the Poincar´e-type inequality for u ∈ W01,A (Ω ),
∥u∥A ≤ C∥∇u∥A i.e. ∥∇u∥A is an equivalent norm on W01,A (Ω ). We give several remarks on these two propositions.
Remark 2.4. (1) In fact, in [7], professor Fan gives Theorems 2.9 and 2.10 also under the Condition (P0 ). By Remark 2.2, we can see that these two conclusions hold without Condition (P0 ). We want to emphasis that the Condition (P0 ) is not compatible with the condition A ∈ N (Ω ) which implies that = 0. limt→0+ A(x,t) t (2) In [7], professor Fan gives Theorem 2.10 also under the following condition exists for all x ∈ Ω and t ∈ [0, +∞), or the following condition is satisfied: (∂A) Either ∂A(x,t) ∂t t ∂A(x,t) ∂t+ lim uniformly for x ∈ Ω , 1 = 0 t→+∞ (A(x, t))1+ N where ∂A(x,t) denotes the right derivative of the convex function A(x, ·) at t. ∂t+ In fact, similar to the proof of Proposition 2.1 in [10], under our condition A, A∗ ∈ N (Ω ) and (P2 ), the Condition (∂A) is automatically satisfied. 3. Differential operators in Musielak–Sobolev spaces and the variational structure Firstly we will give the propositions of differential operators in Musielak–Sobolev spaces. We want to in Remark 3.1. emphasis that some of them are new in our case, e.g. Condition (B) We say that A : Ω × [0, +∞) → [0, +∞) satisfies Condition (A ), if A ∈ N (Ω ), both A and A′ are locally integrable and satisfy (∆2 ) and (A2 ), and A satisfies (P1 )–(P4 ) appeared in Section 2. In the following of this paper we always assume that A ∈ N (Ω ) satisfies (A ). In this case, LA (Ω ), W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are separable, reflexive Banach spaces. if the following conditions (B0 )–(B3 ) We say that a function b : Ω × [0, +∞) → R satisfies Condition (B), are satisfied: (B0 ) b : Ω × [0, +∞) → R is a Carath´eodory function which satisfies Carath´eodory condition, and limt→0+ b(x, t)t = 0 for a.e. x ∈ Ω ; (B1 ) There exists a positive constant b1 such that |b(x, t)|t2 ≤ b1 A(x, t)
for x ∈ Ω and t ∈ [0, +∞);
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(B2 ) There exists a positive constant b2 such that b(x, t)t ≥ b2 a(x, t)
for x ∈ Ω and t ∈ [0, +∞);
(B3 ) b(x, |ξ|)ξ − b(x, |η|)η · (ξ − η) ≥ 0 for x ∈ Ω and ξ, η ∈ RN . Remark 3.1. (1) Our conditions (B1 ) and (B2 ) are stronger than condition (A1,2 ) and (A1,3 ) respectively in [6]. In fact, by Proposition 2.1(1) and (B1 ) |b(x, t)|t ≤ b1
A(x, t) ≤ b1 a(x, t), t
for t > 0,
which implies the condition (A1,2 ) in [6]. By Proposition 2.1(1) and (B2 ) b(x, t)t2 ≥ b2 a(x, t)t ≥ b2 A(x, t), which implies the condition (A1,3 ) in [6]. (2) From (B1 ) and (B2 ), we have an increasing condition on A(x, t): b1
A(x, t) ≤ C|t| b2 ,
∀x ∈ Ω , t ∈ [M, +∞)
for a given M > 0. (3) A variable exponent example satisfying condition (B) is that A(x, t) := |t|p(x) and b(x, t) := |t|p(x)−2 for p ∈ C(Ω ). In this case, we can choose the parameters b1 ≥ 1 and b2 ≤ p1+ , where p+ := supx∈Ω p(x), in assumptions (B1 ) and (B2 ) respectively. The conditions (B0 ) and (B3 ) can be easily verified. Furthermore if p ∈ C 1−0 (Ω ), then A(x, t) := |t|p(x) satisfies condition (A ). We prove only conditions (P3 ) and (P4 ) here. In fact, basic calculation yields that A−1 ∗ (x, s) =
np(x) n−p(x) s np(x) . n − p(x)
Then (P3 ) is followed. For the condition (P4 ), note that there exist constants t0 > 0, ϵ0 > 0 such that |∇x A(x, t)| = |t|p(x) ln |t | ∇p(x)| ≤ C|t|p(x)(1+ϵ0 ) = C(A(x, t))1+ϵ0 , for x ∈ Ω and t ∈ [t0 , T (x)). Since A satisfies (∆2 ), from Proposition 3.1 in [7], it is clear (P4 ) is satisfied. Define φ : W01,A (Ω ) → (W01,A (Ω ))∗ by ⟨φ(u), v⟩ = b(x, |∇u|)∇u · ∇v dx, Ω
∀u, v ∈ W01,A (Ω ).
We can see that the operator φ is the divergence type operator and φ is of variational form, and φ(u) = B ′ (u), where B(u) = B(x, |∇u|) dx, (3.1) Ω
and B(x, t) =
t 0
b(x, s)s ds.
Definition 3.1. Let Y be a real reflexive Banach space. A mapping T : Y → Y ∗ is said to be of type (S+ ) if for any sequence {un } ⊂ X for which un ⇀ u0 weakly in Y and limn→+∞ ⟨T (un ), un − u0 ⟩ ≤ 0, un must converge strongly to u0 in Y .
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By Remark 3.1, we have the following. Proposition 3.1 (See Theorems 2.1 and 2.2 in [6]). Let A ∈ N (Ω ) and b : Ω × [0, +∞) satisfy (A ) and (B) 1,A 1,A ∗ respectively. Then the mapping φ : W0 (Ω ) → (W0 (Ω )) is bounded, continuous, coercive and monotone. If in addition, b(x, t)t is strictly monotone in the second variable, that is, (B3 ) is replaced by (B3 )+ b(x, |ξ|)ξ − b(x, |η|)η · (ξ − η) > 0 for x ∈ Ω and ξ, η ∈ RN with ξ ̸= η, then φ : W01,A (Ω ) → (W01,A (Ω ))∗ is of type (S+ ), and it is a strictly monotone homeomorphism. We say that b : Ω × [0, +∞) satisfies (B) if (B0 ), (B1 ), (B2 ) and (B3 )+ hold. We assume the perturbation f (x, t) is a Carath´eodory function. We introduce the variational framework of problem (1.1). Under some increasing conditions on f about the item u, we observe that (1.1) is the Euler–Lagrange equation associated with the energy functional J (u) := B(x, |∇u|) dx − F (x, u) dx, (3.2) Ω
Ω
t
where F (x, t) = 0 f (x, s) dx. We give the following definition of the weak solution for problem (1.1). Definition 3.2. We say u is a weak solution for problem (1.1), if f (·, u(·)) ∈ L(A∗ ) (Ω ) and b(x, |∇u|)∇u · ∇v dx = f (x, u)v dx, ∀v ∈ W01,A (Ω ). ′
Ω
Ω
4. The existence of weak solutions For simplicity, we make use of the following notations. X denotes Musielak–Sobolev space W01,A (Ω ) with the norm ∥ · ∥ := ∥ · ∥W 1,A (Ω) ; X ∗ denotes the conjugate space for X; LA (Ω ) denotes Musielak space with the norm | · |A ; ⟨·, ·⟩ is the dual pairing on the space X ∗ and X; by → (resp. ⇀) we mean strong (resp. weak) convergence. |Ω | denotes the Lebesgue measure of the set Ω ⊂ RN ; C, C1 , C2 , . . . denote (possibly different) positive constants. Please note that in this section we always assume A ∈ N (Ω ) and b : Ω × [0, +∞) satisfy (A ) and (B) respectively. Let F(u) = Ω F (x, u(x)) dx. Then by the notation in Section 3, J = B − F. If f is independent of u, we have Theorem 4.1. If f (x, u) = f (x) ∈ L(A∗ ) (Ω ), then (1.1) has a unique weak solution. ′
Proof. It is clear that (f, u) := Ω f (x)u dx, ∀u ∈ X, defines a continuous linear functional on X. By Proposition 3.1, (1.1) has a unique weak solution. Next we assume the following conditions on f , (F1) There exists Ψ ∈ N (Ω ) satisfying Ψ ≪ A∗ such that that
∂Ψ ∂s (x, s)
|f (x, t)| ≤ k1 ψ(x, |t|) + h(x), ′
in which k1 is a positive constant and 0 ≤ h ∈ LΨ (Ω );
exists and equals ψ(x, s) for s ≥ 0, such
∀(x, t) ∈ Ω × R,
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(F2) There exist a positive constants θ, k2 and M such that for any t ≥ M the following conditions hold 0 < θF (x, t) ≤ tf (x, t) < k2 A∗ (x, t),
∀x ∈ Ω ,
where θ satisfies θ > bb21 , and b1 , b2 are constants appearing in the assumption (B). (F3) f (x, t) = o(a(x, |t|)), t → 0, uniformly for x ∈ Ω . Lemma 4.1. Under condition (F1), (i) the functional F is sequentially weakly continuous, i.e., un ⇀ u in X implies F(un ) → F(u); (ii) F ′ : X → X ∗ is a completely continuous linear operator. Proof. (i) By (F1) and Proposition 2.3(4) there holds |F(un ) − F(u)| = F (x, un ) − F (x, u) dx Ω un = f (x, t) dt dx Ω u u n |f (x, t)| dt dx ≤ Ω u u n ≤ k1 ψ(x, |t|) + h(x) dt dx Ω u ≤ k1 Ψ (x, |un |) − Ψ (x, |u|) + h(x)un − u dx Ω ≤ k1 Ψ (x, |un |) − Ψ (x, |u|) dx + 2|h|Ψ ′ un − uΨ . Ω
By the Musielak–Sobolev compact embedding theorem we know un → u in LΨ (Ω ), which implies, by Proposition 2.3(4), that |Ψ (x, |un |) − Ψ (x, |u|)| dx → 0. Ω
So the right hand side of the above inequality converges to 0, which implies |F(un ) − F(u)| → 0 as n → +∞. (ii) By (F1), we have |F (x, t)| ≤ C1 Ψ (x, |t|) + h(x)|t|,
∀(x, |t|) ∈ Ω × R.
Then it is easy to see that F ∈ C 1 (X) and F ′ : X → X ∗ defined by ⟨F ′ (u), v⟩ = ⟨F ′ (u), v⟩LΨ ′ ,LΨ = f (x, u(x))v(x) dx,
∀u, v ∈ X ⊂ LΨ (Ω ),
Ω
is continuous (see [6]). By Proposition 2.6 and Theorem 2.10, we have the following embedding sequence i
f (x,·)
′
j
k
X→ − LΨ (Ω ) −−−→ LΨ (Ω ) − → (LΨ (Ω ))∗ − → X ∗, and the following equation F ′ (u) = k ◦ j ◦ f ◦ i(u),
∀u ∈ X,
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in which, i is compact embedding, j is homeomorphic in Proposition 2.6, and k means restriction on X of functionals in (LΨ (Ω ))∗ . Then it is clear that F ′ is completely continuous. By Lemma 4.1, we can see that u ∈ X is the critical point of the functional J if and only if u ∈ X is the weak solution of the problem (1.1). Remark 4.1. Under Assumption (F1), by Lemma 4.1 and Proposition 3.1, we know that J ′ = B ′ − F ′ is of type (S+ ). In the following text, we always assume condition (A∞ ) holds: (A∞ ) There exists a continuous function A∞ : [0, +∞) → [0, +∞), such that A∞ (α) → +∞ as α → +∞ and A(x, αt) ≥ A∞ (α)αA(x, t) for any x ∈ Ω , t ∈ [0, +∞) and α > 0; Under assumption (A∞ ), (Ψ∞ ) and (F1) the following assumptions on A and Ψ will be used, (Ψ∞ ) There exists a nondecreasing continuous function Ψ∞ : [0, +∞) → [0, +∞), such that Ψ∞ (α) → +∞ as α → +∞ and Ψ (x, αt) ≤ Ψ∞ (α)αΨ (x, t) for any x ∈ Ω , t ∈ [0, +∞), α > 0. − ) The function Ψ∞ in assumption (Ψ∞ ) satisfies (Ψ∞ lim
β→+∞
Ψ∞ (Cβ) =0 A∞ (β)
for any given C > 0. (A0 ) There exists a nondecreasing function A0 : [0, +∞) → [0, +∞), such that A0 (α) → 0 as α → 0+ and A(x, αt) ≤ A0 (α)αA(x, t) ∀x ∈ Ω , t ∈ [0, +∞), α > 0. For any given C > 0 and M > 0, there exists a δ > 0 such that A0 (Cβ) < M, A∞ (β)
∀β < δ.
(Ψ0 ) There exists a nondecreasing function Ψ0 : [0, +∞) → [0, +∞), such that Ψ0 (α) → 0 as α → 0+ and Ψ (x, αt) ≤ Ψ0 (α)αΨ (x, t)
and
lim
β→0+
Ψ0 (Cβ) =0 A∞ (β)
for any x ∈ Ω , t ∈ [0, +∞), α > 0, and any given C > 0. Theorem 4.2. Suppose A satisfies condition (A∞ ). Let the Condition (F1) hold and the function Ψ in − assumption (F1) satisfies the assumptions (Ψ∞ ) and (Ψ∞ ). Then the problem (1.1) has a weak solution.
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Proof. By (F1) we have |F (x, t)| ≤ C1 Ψ (x, |t|) + h(x)|t|,
∀(x, t) ∈ Ω × R.
Then J is coercive because of the following inequality J (u) = B(u) − F(u) ≥ B(x, |∇u|) dx − |F (x, u)| dx Ω Ω ≥ b2 A(x, |∇u|) dx − C1 Ψ (x, |u|) dx − 2|h|Ψ ′ |u|Ψ Ω Ω |u| |∇u| dx − C1 dx − C|u|Ψ Ψ x, |u|Ψ = b2 A x, ∥u∥ ∥u∥ |u|Ψ Ω Ω |∇u| |u| dx − C1 Ψ∞ (|u|Ψ )|u|Ψ dx − C|u|Ψ ≥ b2 A∞ (∥u∥)∥u∥ A x, Ψ x, ∥u∥ |u|Ψ Ω Ω = b2 A∞ (∥u∥)∥u∥ − C1 Ψ∞ (|u|Ψ )|u|Ψ − C|u|Ψ ≥ b2 A∞ (∥u∥)∥u∥ − CΨ∞ (C∥u∥)∥u∥ − C∥u∥ = (b2 A∞ (∥u∥) − CΨ∞ (C∥u∥) − C)∥u∥ → +∞, as ∥u∥ → +∞. By Proposition 3.1 and Lemma 4.1, it is easy to verify that J is weakly lower semi-continuous. Then J has a minimum point u in X and u is a weak solution of (1.1), which completes the proof. Lemma 4.2. Suppose A satisfies condition (A∞ ). Then under Assumptions (F1) and (F2), the functional J satisfies the (P.S.) condition. Proof. Suppose that {un } ⊂ X, |J (un )| ≤ C0 for some C0 ∈ R, and J ′ (un ) → 0 in X ∗ as n → ∞. Let c := supn J (un ) and β ∈ ( θ1 , bb21 ) for large n. From Proposition 2.1, Theorem 2.10, (F2) and (A3 ), for big ∥un ∥ we have, c + 1 + ∥un ∥ ≥ J (un ) − β⟨J ′ (un ), un ⟩ 2 = B(x, |∇un |) dx − β b(x, |∇un |)|∇un | dx + βf (x, un )un − F (x, un ) dx Ω Ω Ω ≥ b2 A(x, |∇un |) dx − b1 β A(x, |∇un |) dx + βf (x, un )un − F (x, un ) dx Ω Ω Ω ≥ b2 − b1 β A(x, |∇un |) dx + βθ − 1 F (x, un ) dx Ω Ω ≥ b2 − b1 β A(x, |∇un |) dx Ω |∇un | = b2 − b1 β A x, ∥un ∥ dx ∥un ∥ Ω |∇un | ≥ b2 − b1 β A∞ (∥un ∥)∥un ∥ A x, dx ∥un ∥ Ω = b2 − b1 β A∞ (∥un ∥)∥un ∥. Since b2 − b1 β > 0 and A∞ (t) → +∞ as t → +∞, we obtain the boundedness of {un } in X by the above inequality. Without loss of generality, we assume un ⇀ u, then ⟨J ′ (un ) − J ′ (u), un − u⟩ → 0. Since J ′ is of type (S+ ), we have un → u in X.
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Theorem 4.3. Suppose A satisfies condition (A∞ ) and (A0 ). Let the function Ψ in (F1) satisfies (Ψ0 ) and h(x) ≡ 0 in (F1). Then under Conditions (F1)–(F3), the problem (1.1) has a nontrivial solution. Proof. We will show that the functional J satisfies the geometry condition of the Mountain Pass Theorem. By Lemma 4.2, A satisfies (P.S.) condition in X. By Assumptions (F1) and (F3), there exists some ϵ > 0 small enough such that |F (x, t)| ≤ ϵA(x, |t|) + C(ϵ)Ψ (x, |t|),
∀(x, t) ∈ Ω × [0, +∞).
So by (A0 ) and (Ψ0 ) we have for small ∥u∥ and ϵ small enough, J (u) ≥ b2 A(x, |∇u|) dx − ϵ A(x, |u|) dx − C(ϵ) Ψ (x, |u|) dx Ω Ω Ω |∇u| |u| |u| ≥ b2 A x, ∥u∥ dx − ϵ A x, |u|A dx − C(ϵ) Ψ x, |u|Ψ dx ∥u∥ |u|A |t|Ψ Ω Ω Ω |u| |∇u| dx − ϵA0 (|u|A )|u|A A x, dx ≥ b2 A∞ (∥u∥)∥u∥ A x, ∥u∥ |u|A Ω Ω |u| − C(ϵ)Ψ0 (|u|Ψ )|u|Ψ Ψ x, dx |u|Ψ Ω = b2 A∞ (∥u∥)∥u∥ − ϵA0 (|u|A )|u|A − C(ϵ)Ψ0 (|u|A∗ )|u|A∗ ≥ b2 A∞ (∥u∥)∥u∥ − ϵCA0 (C∥u∥)∥u∥ − CC(ϵ)Ψ0 (C∥u∥)∥u∥ b2 ≥ A∞ (∥u∥)∥u∥ − CC(ϵ)Ψ0 (C∥u∥)∥u∥. 2 Since limβ→0+
Ψ0 (Cβ) A∞ (β)
= 0, there exist r > 0 and δ > 0 such that J (u) ≥ δ > 0 for every ∥u∥ = r.
From the Assumption (F2), (B1 ) and (B2 ), there exist constants C1 , C2 > 0 such that b1
F (x, s) ≥ C1 |s|θ > C1 |s| b2 ≥ C2 A(x, |s|),
for |s| ≥ M, ∀x ∈ Ω .
For w ∈ X \ {0} and big t > 0, in view of the above inequality and θ > bb21 we have J (tw) = B(x, |t∇w|) dx − F (x, tw) dx Ω Ω ≤ b1 A(x, |t∇w|) dx − C1 tθ |w|θ dx − C Ω Ω b1 b1 θ b2 b2 |∇w| dx − C1 t |w|θ dx − C ≤ C3 t Ω
→ −∞,
Ω
as t → +∞.
Obviously we have J (0) = 0, so J satisfies the geometry conditions of the Mountain Pass Theorem in [11]. Then J admits at least one nontrivial critical point which corresponds to the weak solution of (1.1). In the following text, we will use the following assumption on f . (F4) f (x, −t) = −f (x, t), ∀t ∈ R, x ∈ Ω . Thanks to the Assumption (F4), the functional J is even. We can make an application of Fountain theorem and Dual Fountain theorem to get infinitely many solutions to (1.1). Theorem 4.4. Suppose A satisfies condition (A∞ ). Let (F1), (F2) and (F4) hold, the function Ψ in assumption (F1) satisfies assumption (Ψ∞ ) and h(x) ≡ 0 in (F1). Then (1.1) has a sequence of weak solutions {±uk }∞ k=1 such that J (±uk ) → +∞ as k → +∞.
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327
We will use Fountain theorem to prove Theorem 4.4. By Propositions 2.4 and 2.6, X is a reflexive and separable Banach space, there exist {ej } ⊂ X and {e∗j } ⊂ X ∗ such that X = span{ej : j = 1, 2, . . .},
X ∗ = span{e∗j : j = 1, 2, . . .}
in which ⟨ei , e∗j ⟩
1, = 0,
i = j, i ̸= j.
Denote Xj = span{ej }, Yk = ⊕kj=1 Xj , Zk = ⊕∞ j=k Xj . Lemma 4.3. If Ψ ≪ A∗ , denote Ψ (x, |u|) dx : ∥u∥ = 1, u ∈ Zk , βk = sup Ω
then limk→+∞ βk = 0. Proof. It is obvious that 0 < βk+1 < βk . Then there exists a β > 0 such that βk → β as k → +∞. By the definition of βk , for any k > 0, there exists uk ∈ Zk , such that ∥uk ∥ = 1 and 0 ≤ β − |uk |Ψ < k1 . Since ∥uk ∥ = 1, there exists a subsequence of {uk } (still denoted by {uk }) such that uk ⇀ u in W 1,A (Ω ), and by the definition of Zk we have ⟨e∗j , u⟩ = lim ⟨e∗j , u⟩ = 0, k→+∞
∀j = 1, 2, . . . .
Then u = 0, which implies that uk ⇀ 0 in W 1,A (Ω ). By Theorem 2.10 the embedding W 1,A (Ω ) ↩→↩→ LΨ (Ω ) is compact, then uk → 0 in LΨ (Ω ), which implies that βk → 0 by Proposition 2.3. The proof of the lemma is completed. Lemma 4.4 (Fountain Theorem, See [11]). Assume (M1) X is a Banach space, J ∈ C 1 (X, R) is an even functional. If for each k ∈ N, there exists ρk > rk > 0 such that (M2) inf u∈Zk ,∥u∥=rk J (u) → +∞ as k → +∞. (M3) maxu∈Yk ,∥u∥=ρk J (u) ≤ 0 (M4) J satisfies (P.S.)c condition for every c > 0. Then J admits a sequence of critical values tending to +∞. The proof of Theorem 4.4. By the assumption (F4), F is even, which implies J = B−F is also even. Further more, by Lemma 4.2, A satisfies the (P.S.)c condition. We need only to prove that there exists ρk > rk > 0 such that condition (M2) and (M3) in Lemma 4.4 hold. (M2). Let u ∈ Zk . By (F1), Proposition 2.3 and the definition of βk , we have J (u) = B(u) − F(u) ≥ B(x, |∇u|) dx − |F (x, u)| dx Ω
Ω
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1 A(x, |∇u|) dx − C Ψ (x, |u|) dx − 2|h|Ψ ′ |u|Ψ Ω Ω |u| |∇u| 1 dx − C dx − C|u|Ψ Ψ x, ∥u∥ = b2 A x, ∥u∥ ∥u∥ ∥u∥ Ω Ω |u| |∇u| 1 Ψ∞ (∥u∥)∥u∥ 2 ∥u∥ dx − C Ψ x, dx − C ≥ b2 A∞ (∥u∥)∥u∥ A x, ∥u∥ ∥u∥ Ω Ω ≥ b2
1 Ψ∞ (∥u∥)∥u∥βk − C 2 ∥u∥. ≥ b2 A∞ (∥u∥)∥u∥ − C
(4.1)
By Assumption (F2), (B1 ) and (B2 ), there exist constants C1 , C2 > 0 such that b1
F (x, s) ≥ C1 |s|θ > C1 |s| b2 ≥ C2 A(x, |s|),
for |s| ≥ M, ∀x ∈ Ω .
(4.2)
At the same time from assumption (F1), (A∞ ) and (Ψ∞ ), for |s| = tM we have CΨ∞ (t)tΨ (x, M ) + C ≥ CΨ (x, |s|) + C ≥ F (x, s), A(x, |s|) ≥ A∞ (t)tA(x, M ). Then assumption θ >
b1 b2 ,
(4.3)
(4.2) and (4.3) implies lim
t→+∞
A∞ (t) = 0. Ψ∞ (t)
Since βk → 0 as k → +∞, by the continuity of the function A∞ and Ψ∞ , for big k there exists rk (= rk (βk ) → +∞ as k → +∞) such that 1 1 βk Ψ (rk ) + C 2 . b2 A∞ (rk ) = C 2 Set ∥u∥ = rk in the inequality (4.1) we have 1 Ψ∞ (∥u∥)∥u∥βk − C 2 ∥u∥ J (u) ≥ b2 A∞ (∥u∥)∥u∥ − C 1 = b2 A∞ (rk )rk → +∞ as k → +∞. 2 (M3). By (4.2), for any w ∈ Yk with ∥w∥ = 1 and ρk = t > rk , we have J (tw) = B(x, |t∇w|) dx − F (x, tw) dx Ω Ω θ ≤ b1 A(x, |t∇w|) dx − C1 t |w|θ dx − C Ω Ω b1 b1 θ b2 b2 ≤ C3 t |∇w| dx − C1 t |w|θ dx − C Ω b1 b2
= C3 t ∥w∥
Ω b1 b2
b 1, 1 W b2
θ
(Ω)
− C1 t |w|θLθ (Ω) dx − C.
Since all norms in finite dimensional space Yk are equivalent and θ > The conclusion of Theorem 4.4 is obtained by Lemma 4.4.
p1 p2 ,
we have J (tw) → −∞.
Acknowledgments The authors are very grateful to the reviewers for their valued comments. This research is supported by the National Natural Science Foundation of China (NSFC 11471147) and Fundamental Research Funds for the Central Universities (lzujbky-2014-25).
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References [1] G. Dai, D. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009) 704–710. [2] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. [3] X.L. Fan, D. Zhao, On the generalized Orlicz–Sobolev space wk,p(x) (ω), J. Gansu Educ. Coll. 12 (1) (1998) 1–6. [4] X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x)-growth conditions, Nonlinear Anal. 39 (2000) 807–816. [5] A. Benkirane, M. Sidi El Vally, An existence result for nonlinear elliptic equations in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. 20 (1) (2013) 1–187. [6] X.L. Fan, Differential equations of divergence form in Musielak–Sobolev spaces and sub-supersolution method, J. Math. Anal. Appl. 386 (2012) 593–604. [7] X.L. Fan, An embedding theorem for Musielak–Sobolev spaces, Nonlinear Anal. 75 (2012) 1959–1971. [8] J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Math., vol. 1034, Springer-Verlag, Berlin, 1983. [9] R. Adams, Sobolev Spaces, Acad. Press, New York, 1975. [10] M. Garc´ıa-Huidobro, V.K. Le, R. Man´ asevich, K. Schmitt, On principle eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999) 207–225. [11] M. Willem, Minimax Theorems, Birkh¨ auser, Boston, 1996.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
Linear Algebra and its Applications Nonlinear Analysis: Real World Applications www.elsevier.com/locate/laa www.elsevier.com/locate/nonrwa
eigenvalue Solutions for aInverse quasilinear ellipticproblem equationofinJacobi matrix withspaces mixed data Musielak–Sobolev YingZhao Wei 1 Duchao Liu ∗ , Peihao School of Mathematics Department and Statistics, Lanzhou University, 730000, PR China and Astronautics, of Mathematics, NanjingLanzhou University of Aeronautics 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: Critical point theory is used to show the existence of weak solutions to a quasilinear Article history: Received 16 August 2014 In thisunder paper, the inverse eigenvalue of reconstructing elliptic differential equation the functional frameworkproblem of the Musielak–Sobolev Received 16 January 2014 Received in revised form 12 June a Jacobi matrix from its eigenvalues, its leading principal spaces in a bounded smooth domain with Dirichlet boundary condition. Accepted 20 September 2014 2015 submatrix and part© of theElsevier eigenvalues of rights its submatrix 2015 Ltd. All reserved. Accepted 23 June 2015 Available online 22 October 2014 is considered. The necessary and sufficient conditions for Submitted by Y. Wei Dedicated to Professor Xianling Fan MSC: on the occasion of his 70th birthday 15A18 15A57 Keywords: Musielak–Sobolev space Keywords: Critical point theory Jacobi matrix Mountain Pass type solution Eigenvalue Fountain theorem Inverse problem Submatrix
the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given. © 2014 Published by Elsevier Inc.
1. Introduction In this paper we deal with the existence of weak solutions to a differential equation in the Musielak–Sobolev spaces of the form −div b(x, |∇u|)∇u = f (x, u) in Ω , (1.1) u=0 on ∂Ω , where Ω ⊂ RN is a bounded smooth domain. In the study of nonlinear differential equations, it is well known that more general functional space can handle differential equations with more nonlinearities. If we want to study a general form of differential E-mail address: [email protected]. 1 Tel.: +86 13914485239. equations, it is very important to find a proper functional space corresponding to their solutions. Differential equations in variable exponent Sobolev spaces and Orlicz–Sobolev spaces have been http://dx.doi.org/10.1016/j.laa.2014.09.031 Published by Elsevier Inc.best knowledge, however, differential equations in studied extensively 0024-3795/© in recent 2014 years, see [1–4]. To our ∗ Corresponding author. Tel.: +86 13893289235; fax: +86 09318912481. E-mail addresses: [email protected] (D. Liu), [email protected] (P. Zhao).
http://dx.doi.org/10.1016/j.nonrwa.2015.06.002 1468-1218/© 2015 Elsevier Ltd. All rights reserved.
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Musielak–Sobolev spaces have been studied little. In the research of the paper [5], Benkirane and Sidi give an existence result for nonlinear elliptic equations in Musielak–Sobolev spaces, and in this paper the authors give an embedding theorem in Musielak–Sobolev spaces. In two recent papers (see [6,7]), Fan gives some properties about this kind of functional space. It is possible to make an application of those properties to give some existence of solutions to the elliptic equations with much more nonlinearities. Our aim in this paper is to study the existence of solutions to a kind of elliptic differential equation under the functional frame work of Musielak–Sobolev spaces, which is a more general case of the variable exponent Sobolev spaces and Orlicz–Sobolev spaces. The method we used here is the variational method. The paper is organized as follows. In Section 2, for the readers convenience we recall some definitions and properties about Musielak–Orlicz–Sobolev spaces. In Section 3, we give some basic properties of the differential operator in Musielak–Sobolev spaces corresponding to the equation. In Section 4, we give several existence results for the weak solutions to our problem (1.1). 2. The Musielak–Orlicz–Sobolev spaces In this section, we list some definitions and propositions (some of them are new in our case, e.g. Proposition 2.3(4), Remark 2.2, Remark 2.4 for Theorem 2.10) about Musielak–Orlicz–Sobolev spaces for the readers to refer. Firstly, we give the definition of “N -function” and “generalized N -function” as follows. Definition 2.1. A function A : R → [0, +∞) is called an N -function, denoted by A ∈ N , if A is even and convex, A(0) = 0, 0 < A(t) ∈ C 0 for t ̸= 0, and the following conditions hold lim
t→0+
A(t) =0 t
and
lim
t→+∞
A(t) = +∞. t
A function A : Ω × R → [0, +∞) is called a generalized N -function, denoted by A ∈ N (Ω ), if for each t ∈ [0, +∞), the function A(·, t) is measurable, and for a.e. x ∈ Ω , we have A(x, ·) ∈ N . Let A ∈ N (Ω ), the Musielak–Orlicz space LA (Ω ) is defined by |u(x)| LA (Ω ) := u : u is a measurable real function, and ∃λ > 0 such that A x, dx < +∞ λ Ω with the (Luxemburg) norm
∥u∥LA (Ω)
|u(x)| = ∥u∥A := inf λ > 0 : A x, λ Ω
dx ≤ 1 .
The Musielak–Sobolev space W 1,A (Ω ) can be defined by W 1,A (Ω ) := {u ∈ LA (Ω ) : |∇u| ∈ LA (Ω )} with the norm ∥u∥W 1,A (Ω) = ∥u∥1,A := ∥u∥A + ∥∇u∥A , where ∥∇u∥A := ∥ |∇u| ∥A . Definition 2.2. We say that A satisfies Condition (∆2 ), if there exist a positive constant K > 0 and a nonnegative function h ∈ L1 (Ω ) such that A(x, 2t) ≤ KA(x, t) + h(x)
for x ∈ Ω and t ∈ R.
A is called locally integrable if A(·, t0 ) ∈ L1 (Ω ) for every t0 > 0. For x ∈ Ω and t ≥ 0, we denote by a(x, t) the right-hand derivative of A(x, ·) at t, at the same time define |t| a(x, t) = −a(x, −t). Then A(x, t) = 0 a(x, s) ds for x ∈ Ω and t ∈ R.
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Define A′ : Ω × R → [0, +∞) by A′ (x, s) = sup st − A(x, t)
for x ∈ Ω and s ∈ R.
t∈R
A′ is called the complementary function to A in the sense of Young. It is well known that A′ ∈ N (Ω ) and A is also the complementary function to A′ . ′ For x ∈ Ω and s ≥ 0, we denote by a−1 + (x, s) the right-hand derivative of A (x, ·) at s at the same time −1 −1 define a+ (x, s) = −a+ (x, −s) for x ∈ Ω and s ≤ 0. Then for x ∈ Ω and s ≥ 0, we have a−1 + (x, s) = sup{t ≥ 0 : a(x, t) ≤ s} = inf{t > 0 : a(x, t) > s}. Proposition 2.1. Let A ∈ N (Ω ). Then the following assertions hold. (1) A(x, t) ≤ a(x, t)t ≤ A(x, 2t) for x ∈ Ω and t ∈ R; (2) A and A′ satisfy the Young inequality st ≤ A(x, t) + A′ (x, s) for x ∈ Ω and s, t ∈ R and the equality holds if s = a(x, t) or t = a−1 + (x, s). Let A, B ∈ N (Ω ). We say that A is weaker than B, denoted by A 4 B, if there exist positive constants K1 , K2 and a nonnegative function h ∈ L1 (Ω ) such that A(x, t) ≤ K1 B(x, K2 t) + h(x)
for x ∈ Ω and t ∈ [0, +∞).
(2.1) ′
′
Proposition 2.2. Let A, B ∈ N (Ω ) and A 4 B. Then B ′ 4 A′ , LA (Ω ) ↩→ LB (Ω ) and LB (Ω ) ↩→ LA (Ω ). Proposition 2.3. Let A ∈ N (Ω ) satisfy (∆2 ). Then the following assertions hold, (1) (2) (3) (4) (5)
LA (Ω ) = {u : u is a measurable function, and Ω A(x, |u(x)|) dx < +∞}; A(x, |u|) dx < 1(resp. = 1; > 1) ⇐⇒ ∥u∥A < 1(resp. = 1; > 1), where u ∈ LA (Ω ); Ω A(x, |un |) dx → 0(resp. 1; +∞) ⇐⇒ ∥un ∥A → 0(resp. 1; +∞), where {un } ∈ LA (Ω ); Ω A un → u in L (Ω ) =⇒ Ω A(x, |un |) dx − A(x, |u|) dx → 0 as n → ∞; If A′ also satisfies (∆2 ), then u(x)v(x) dx ≤ 2∥u∥A ∥v∥A′ ,
′
∀u ∈ LA (Ω ), v ∈ LA (Ω );
Ω
′
(6) a(·, |u(·)|) ∈ LA (Ω ) for every u ∈ LA (Ω ). Proof. The readers can find the proofs of (1–3) and (5–6) in [6]. For the item (4), in the Ref. [6], professor Fan only proves that “(4′ ) un → u in LA (Ω ) =⇒ Ω A(x, |un |) dx → Ω A(x, |u|) dx as n → ∞”, we can prove that this conclusion is equivalent to the conclusion in Proposition 2.3(4). It is trivial that (4) =⇒ (4′ ) is right. We can prove (4′ ) =⇒ (4). In fact, if we set un , for x ∈ {|un | ≥ |u|}, + un = u, for x ∈ {|un | < |u|}; u, for x ∈ {|un | ≥ |u|}, u− n = un , for x ∈ {|un | < |u|},
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A then u± n → u in L (Ω ). By the conclusion in Ref. [6], we have A(x, |u+ n |) − A(x, |u|) dx → 0,
(2.2)
Ω
and
A(x, |u− n |) − A(x, |u|) dx → 0,
(2.3)
Ω
as n → +∞. By taking (2.2)–(2.3), we can get − A(x, |un |) − A(x, |u|) dx = A(x, |u+ n |) dx − A(x, |un |) dx → 0, Ω
as n → +∞.
Ω
Proposition 2.4. Let A ∈ N (Ω ) be locally integrable. Then (LA (Ω ), ∥ · ∥) is a separable Banach space. The following assumptions will be used. (A1 ) inf x∈Ω A(x, 1) = c1 > 0; (A2 ) For every t0 > 0, there exists c = c(t0 ) > 0 such that A(x, t) ≥ c and t
A′ (x, t) ≥ c, ∀t ≥ t0 , x ∈ Ω . t
Remark 2.1. Obviously, (A2 ) ⇒ (A1 ). If A(x, t) = A(t) is an N -function, then (A1 ) and (A2 ) hold automatically, and A is automatically locally integrable. Proposition 2.5. If A ∈ N (Ω ) satisfies (A1 ), then LA (Ω ) ↩→ L1 (Ω ) and W 1,A (Ω ) ↩→ W 1,1 (Ω ). Proposition 2.6 (See [6]). Let A ∈ N (Ω ), both A and A′ be locally integrable and satisfy ∆2 and (A2 ). Then ′ the space LA (Ω ) is reflexive, and the mapping J : LA (Ω ) → (LA (Ω ))∗ defined by ′ ⟨J(v), w⟩ = v(x)w(x) dx, ∀v ∈ LA (Ω ), w ∈ LA (Ω ) Ω
is a linear isomorphism and ∥J(v)∥(LA (Ω))∗ ≤ 2∥v∥LA′ (Ω) . Let A ∈ N (Ω ) be locally integrable. We will denote W01,A (Ω ) := C0∞ (Ω ) D01,A (Ω ) := C0∞ (Ω )
∥ · ∥W 1,A (Ω)
∥∇ · ∥LA (Ω)
.
In the case ∥∇u∥A is an equivalent norm in W01,A (Ω ), W01,A (Ω ) = D01,A (Ω ). Proposition 2.7 (See [6]). Let A ∈ N (Ω ) be locally integrable and satisfy (A1 ). Then (1) the spaces W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are separable Banach spaces, and W01,A (Ω ) ↩→ W 1,A (Ω ) ↩→ W 1,1 (Ω ) D01,A (Ω ) ↩→ D01,1 (Ω ) = W01,1 (Ω );
(2) the spaces W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are reflexive provided LA (Ω ) is reflexive.
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Proposition 2.8. Let A, B ∈ N (Ω ) and A be locally integrable. If there is a compact embedding W 1,A (Ω ) ↩→↩→ LB (Ω ) and A 4 B, then there holds the following Poincar´e inequality ∥u∥A ≤ c∥∇u∥A ,
∀u ∈ W01,A (Ω ),
which implies that ∥∇ · ∥A is an equivalent norm in W01,A (Ω ) and W01,A (Ω ) = D01,A (Ω ). The following assumptions will be used. (P1 ) Ω ⊂ RN (N ≥ 2) is a bounded domain with the cone property, and A ∈ N (Ω ); (P2 ) A : Ω × R → [0, +∞) is continuous and A(x, t) ∈ (0, +∞) for x ∈ Ω and t ∈ (0, +∞). Let A satisfy (P1 ) and (P2 ). Define A1 : Ω × [0, +∞) → [0, +∞) by A(x, 1)|t|, for x ∈ Ω and |t| ∈ [0, 1], A1 (x, t) = A(x, t), for x ∈ Ω and |t| > 1. Remark 2.2. Since Ω is bounded, we can see (see [8,7]) that LA (Ω ) = LA1 (Ω ) and W 1,A (Ω ) = W 1,A1 (Ω ). Sometimes, in the study of embeddings of W 1,A (Ω ), it is much more convenient to consider A1 instead of A. In other words, we may assume that A satisfies the following condition which is not essential because Ω is bounded. (P0 ) A(x, t) = A(x, 1)|t| for x ∈ Ω and |t| ∈ [0, 1). Under assumptions (P1 ) and (P2 ), for each x ∈ Ω we can define A∗ (x, t) for x ∈ Ω and t ≥ 0 as in [7], and define A∗ (x, t) = A∗ (x, −t) for x ∈ Ω and t ≤ 0. We call A∗ the Sobolev conjugate function of A. Set 1 lims→+∞ A−1 ∗ (x, s) = T (x), then A∗ (x, ·) ∈ C (0, T (x)). Furthermore for A ∈ N (Ω ) and T (x) = +∞ for any x ∈ Ω , it is known that A∗ ∈ N (Ω ) (see in [9]). Let X be a metric space and f : X → (−∞, +∞] be an extended real-valued function. For x ∈ X with f (x) ∈ R, the continuity of f at x is well defined. For x ∈ X with f (x) = +∞, we say that f is continuous at x if given any M > 0, there exists a neighborhood U of x such that f (y) > M for all y ∈ U . We say that f : X → (−∞, +∞] is continuous on X if f is continuous at every x ∈ X. Define Dom(f ) = {x ∈ X : f (x) ∈ R} and denote by C 1−0 (X) the set of all locally Lipschitz continuous real-valued functions defined on X. The following assumptions will also be used. (P3 ) T : Ω → [0, +∞] is continuous on Ω and T ∈ C 1−0 (Dom(T )); (P4 ) A∗ ∈ C 1−0 (Dom(A∗ )) and there exist positive constants δ0 < that |∇x A∗ (x, t)| ≤ C0 (A∗ (x, t))1+δ0 ,
1 N,
C0 and 0 < t0 < minx∈Ω T (x) such
j = 1, . . . , N,
for x ∈ Ω and |t| ∈ [t0 , T (x)) provided ∇x A∗ (x, t) exists. Let A, B ∈ N (Ω ). We say that A ≪ B if, for any k > 0, lim
t→+∞
A(x, kt) =0 B(x, t)
uniformly for x ∈ Ω .
Remark 2.3. Suppose that A, B ∈ N (Ω ), then A ≪ B ⇒ A 4 B. We give two embedding theorems for Musielak–Sobolev spaces developed by Fan in [7] with some suitable changes (see Remark 2.4) in order to be more convenient to make an application.
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Theorem 2.9. Let (P1 )–(P4 ) hold. Then (i) There is a continuous embedding W 1,A (Ω ) ↩→ LA∗ (Ω ); (ii) Suppose that B ∈ N (Ω ), B : Ω × [0, +∞) → [0, +∞) is continuous, and B(x, t) ∈ (0, +∞) for x ∈ Ω and t ∈ (0, +∞). If B ≪ A∗ , then there is a compact embedding W 1,A (Ω ) ↩→↩→ LB (Ω ). By Theorem 2.9, Remark 2.3 and Proposition 2.8, we have the following. Theorem 2.10. Let (P1 )–(P4 ) hold and furthermore, A, A∗ ∈ N (Ω ). Then (i) A ≪ A∗ , and there is a compact embedding W 1,A (Ω ) ↩→↩→ LA (Ω ); (ii) there holds the Poincar´e-type inequality for u ∈ W01,A (Ω ),
∥u∥A ≤ C∥∇u∥A i.e. ∥∇u∥A is an equivalent norm on W01,A (Ω ). We give several remarks on these two propositions.
Remark 2.4. (1) In fact, in [7], professor Fan gives Theorems 2.9 and 2.10 also under the Condition (P0 ). By Remark 2.2, we can see that these two conclusions hold without Condition (P0 ). We want to emphasis that the Condition (P0 ) is not compatible with the condition A ∈ N (Ω ) which implies that = 0. limt→0+ A(x,t) t (2) In [7], professor Fan gives Theorem 2.10 also under the following condition exists for all x ∈ Ω and t ∈ [0, +∞), or the following condition is satisfied: (∂A) Either ∂A(x,t) ∂t t ∂A(x,t) ∂t+ lim uniformly for x ∈ Ω , 1 = 0 t→+∞ (A(x, t))1+ N where ∂A(x,t) denotes the right derivative of the convex function A(x, ·) at t. ∂t+ In fact, similar to the proof of Proposition 2.1 in [10], under our condition A, A∗ ∈ N (Ω ) and (P2 ), the Condition (∂A) is automatically satisfied. 3. Differential operators in Musielak–Sobolev spaces and the variational structure Firstly we will give the propositions of differential operators in Musielak–Sobolev spaces. We want to in Remark 3.1. emphasis that some of them are new in our case, e.g. Condition (B) We say that A : Ω × [0, +∞) → [0, +∞) satisfies Condition (A ), if A ∈ N (Ω ), both A and A′ are locally integrable and satisfy (∆2 ) and (A2 ), and A satisfies (P1 )–(P4 ) appeared in Section 2. In the following of this paper we always assume that A ∈ N (Ω ) satisfies (A ). In this case, LA (Ω ), W 1,A (Ω ), W01,A (Ω ) and D01,A (Ω ) are separable, reflexive Banach spaces. if the following conditions (B0 )–(B3 ) We say that a function b : Ω × [0, +∞) → R satisfies Condition (B), are satisfied: (B0 ) b : Ω × [0, +∞) → R is a Carath´eodory function which satisfies Carath´eodory condition, and limt→0+ b(x, t)t = 0 for a.e. x ∈ Ω ; (B1 ) There exists a positive constant b1 such that |b(x, t)|t2 ≤ b1 A(x, t)
for x ∈ Ω and t ∈ [0, +∞);
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(B2 ) There exists a positive constant b2 such that b(x, t)t ≥ b2 a(x, t)
for x ∈ Ω and t ∈ [0, +∞);
(B3 ) b(x, |ξ|)ξ − b(x, |η|)η · (ξ − η) ≥ 0 for x ∈ Ω and ξ, η ∈ RN . Remark 3.1. (1) Our conditions (B1 ) and (B2 ) are stronger than condition (A1,2 ) and (A1,3 ) respectively in [6]. In fact, by Proposition 2.1(1) and (B1 ) |b(x, t)|t ≤ b1
A(x, t) ≤ b1 a(x, t), t
for t > 0,
which implies the condition (A1,2 ) in [6]. By Proposition 2.1(1) and (B2 ) b(x, t)t2 ≥ b2 a(x, t)t ≥ b2 A(x, t), which implies the condition (A1,3 ) in [6]. (2) From (B1 ) and (B2 ), we have an increasing condition on A(x, t): b1
A(x, t) ≤ C|t| b2 ,
∀x ∈ Ω , t ∈ [M, +∞)
for a given M > 0. (3) A variable exponent example satisfying condition (B) is that A(x, t) := |t|p(x) and b(x, t) := |t|p(x)−2 for p ∈ C(Ω ). In this case, we can choose the parameters b1 ≥ 1 and b2 ≤ p1+ , where p+ := supx∈Ω p(x), in assumptions (B1 ) and (B2 ) respectively. The conditions (B0 ) and (B3 ) can be easily verified. Furthermore if p ∈ C 1−0 (Ω ), then A(x, t) := |t|p(x) satisfies condition (A ). We prove only conditions (P3 ) and (P4 ) here. In fact, basic calculation yields that A−1 ∗ (x, s) =
np(x) n−p(x) s np(x) . n − p(x)
Then (P3 ) is followed. For the condition (P4 ), note that there exist constants t0 > 0, ϵ0 > 0 such that |∇x A(x, t)| = |t|p(x) ln |t | ∇p(x)| ≤ C|t|p(x)(1+ϵ0 ) = C(A(x, t))1+ϵ0 , for x ∈ Ω and t ∈ [t0 , T (x)). Since A satisfies (∆2 ), from Proposition 3.1 in [7], it is clear (P4 ) is satisfied. Define φ : W01,A (Ω ) → (W01,A (Ω ))∗ by ⟨φ(u), v⟩ = b(x, |∇u|)∇u · ∇v dx, Ω
∀u, v ∈ W01,A (Ω ).
We can see that the operator φ is the divergence type operator and φ is of variational form, and φ(u) = B ′ (u), where B(u) = B(x, |∇u|) dx, (3.1) Ω
and B(x, t) =
t 0
b(x, s)s ds.
Definition 3.1. Let Y be a real reflexive Banach space. A mapping T : Y → Y ∗ is said to be of type (S+ ) if for any sequence {un } ⊂ X for which un ⇀ u0 weakly in Y and limn→+∞ ⟨T (un ), un − u0 ⟩ ≤ 0, un must converge strongly to u0 in Y .
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By Remark 3.1, we have the following. Proposition 3.1 (See Theorems 2.1 and 2.2 in [6]). Let A ∈ N (Ω ) and b : Ω × [0, +∞) satisfy (A ) and (B) 1,A 1,A ∗ respectively. Then the mapping φ : W0 (Ω ) → (W0 (Ω )) is bounded, continuous, coercive and monotone. If in addition, b(x, t)t is strictly monotone in the second variable, that is, (B3 ) is replaced by (B3 )+ b(x, |ξ|)ξ − b(x, |η|)η · (ξ − η) > 0 for x ∈ Ω and ξ, η ∈ RN with ξ ̸= η, then φ : W01,A (Ω ) → (W01,A (Ω ))∗ is of type (S+ ), and it is a strictly monotone homeomorphism. We say that b : Ω × [0, +∞) satisfies (B) if (B0 ), (B1 ), (B2 ) and (B3 )+ hold. We assume the perturbation f (x, t) is a Carath´eodory function. We introduce the variational framework of problem (1.1). Under some increasing conditions on f about the item u, we observe that (1.1) is the Euler–Lagrange equation associated with the energy functional J (u) := B(x, |∇u|) dx − F (x, u) dx, (3.2) Ω
Ω
t
where F (x, t) = 0 f (x, s) dx. We give the following definition of the weak solution for problem (1.1). Definition 3.2. We say u is a weak solution for problem (1.1), if f (·, u(·)) ∈ L(A∗ ) (Ω ) and b(x, |∇u|)∇u · ∇v dx = f (x, u)v dx, ∀v ∈ W01,A (Ω ). ′
Ω
Ω
4. The existence of weak solutions For simplicity, we make use of the following notations. X denotes Musielak–Sobolev space W01,A (Ω ) with the norm ∥ · ∥ := ∥ · ∥W 1,A (Ω) ; X ∗ denotes the conjugate space for X; LA (Ω ) denotes Musielak space with the norm | · |A ; ⟨·, ·⟩ is the dual pairing on the space X ∗ and X; by → (resp. ⇀) we mean strong (resp. weak) convergence. |Ω | denotes the Lebesgue measure of the set Ω ⊂ RN ; C, C1 , C2 , . . . denote (possibly different) positive constants. Please note that in this section we always assume A ∈ N (Ω ) and b : Ω × [0, +∞) satisfy (A ) and (B) respectively. Let F(u) = Ω F (x, u(x)) dx. Then by the notation in Section 3, J = B − F. If f is independent of u, we have Theorem 4.1. If f (x, u) = f (x) ∈ L(A∗ ) (Ω ), then (1.1) has a unique weak solution. ′
Proof. It is clear that (f, u) := Ω f (x)u dx, ∀u ∈ X, defines a continuous linear functional on X. By Proposition 3.1, (1.1) has a unique weak solution. Next we assume the following conditions on f , (F1) There exists Ψ ∈ N (Ω ) satisfying Ψ ≪ A∗ such that that
∂Ψ ∂s (x, s)
|f (x, t)| ≤ k1 ψ(x, |t|) + h(x), ′
in which k1 is a positive constant and 0 ≤ h ∈ LΨ (Ω );
exists and equals ψ(x, s) for s ≥ 0, such
∀(x, t) ∈ Ω × R,
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(F2) There exist a positive constants θ, k2 and M such that for any t ≥ M the following conditions hold 0 < θF (x, t) ≤ tf (x, t) < k2 A∗ (x, t),
∀x ∈ Ω ,
where θ satisfies θ > bb21 , and b1 , b2 are constants appearing in the assumption (B). (F3) f (x, t) = o(a(x, |t|)), t → 0, uniformly for x ∈ Ω . Lemma 4.1. Under condition (F1), (i) the functional F is sequentially weakly continuous, i.e., un ⇀ u in X implies F(un ) → F(u); (ii) F ′ : X → X ∗ is a completely continuous linear operator. Proof. (i) By (F1) and Proposition 2.3(4) there holds |F(un ) − F(u)| = F (x, un ) − F (x, u) dx Ω un = f (x, t) dt dx Ω u u n |f (x, t)| dt dx ≤ Ω u u n ≤ k1 ψ(x, |t|) + h(x) dt dx Ω u ≤ k1 Ψ (x, |un |) − Ψ (x, |u|) + h(x)un − u dx Ω ≤ k1 Ψ (x, |un |) − Ψ (x, |u|) dx + 2|h|Ψ ′ un − uΨ . Ω
By the Musielak–Sobolev compact embedding theorem we know un → u in LΨ (Ω ), which implies, by Proposition 2.3(4), that |Ψ (x, |un |) − Ψ (x, |u|)| dx → 0. Ω
So the right hand side of the above inequality converges to 0, which implies |F(un ) − F(u)| → 0 as n → +∞. (ii) By (F1), we have |F (x, t)| ≤ C1 Ψ (x, |t|) + h(x)|t|,
∀(x, |t|) ∈ Ω × R.
Then it is easy to see that F ∈ C 1 (X) and F ′ : X → X ∗ defined by ⟨F ′ (u), v⟩ = ⟨F ′ (u), v⟩LΨ ′ ,LΨ = f (x, u(x))v(x) dx,
∀u, v ∈ X ⊂ LΨ (Ω ),
Ω
is continuous (see [6]). By Proposition 2.6 and Theorem 2.10, we have the following embedding sequence i
f (x,·)
′
j
k
X→ − LΨ (Ω ) −−−→ LΨ (Ω ) − → (LΨ (Ω ))∗ − → X ∗, and the following equation F ′ (u) = k ◦ j ◦ f ◦ i(u),
∀u ∈ X,
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in which, i is compact embedding, j is homeomorphic in Proposition 2.6, and k means restriction on X of functionals in (LΨ (Ω ))∗ . Then it is clear that F ′ is completely continuous. By Lemma 4.1, we can see that u ∈ X is the critical point of the functional J if and only if u ∈ X is the weak solution of the problem (1.1). Remark 4.1. Under Assumption (F1), by Lemma 4.1 and Proposition 3.1, we know that J ′ = B ′ − F ′ is of type (S+ ). In the following text, we always assume condition (A∞ ) holds: (A∞ ) There exists a continuous function A∞ : [0, +∞) → [0, +∞), such that A∞ (α) → +∞ as α → +∞ and A(x, αt) ≥ A∞ (α)αA(x, t) for any x ∈ Ω , t ∈ [0, +∞) and α > 0; Under assumption (A∞ ), (Ψ∞ ) and (F1) the following assumptions on A and Ψ will be used, (Ψ∞ ) There exists a nondecreasing continuous function Ψ∞ : [0, +∞) → [0, +∞), such that Ψ∞ (α) → +∞ as α → +∞ and Ψ (x, αt) ≤ Ψ∞ (α)αΨ (x, t) for any x ∈ Ω , t ∈ [0, +∞), α > 0. − ) The function Ψ∞ in assumption (Ψ∞ ) satisfies (Ψ∞ lim
β→+∞
Ψ∞ (Cβ) =0 A∞ (β)
for any given C > 0. (A0 ) There exists a nondecreasing function A0 : [0, +∞) → [0, +∞), such that A0 (α) → 0 as α → 0+ and A(x, αt) ≤ A0 (α)αA(x, t) ∀x ∈ Ω , t ∈ [0, +∞), α > 0. For any given C > 0 and M > 0, there exists a δ > 0 such that A0 (Cβ) < M, A∞ (β)
∀β < δ.
(Ψ0 ) There exists a nondecreasing function Ψ0 : [0, +∞) → [0, +∞), such that Ψ0 (α) → 0 as α → 0+ and Ψ (x, αt) ≤ Ψ0 (α)αΨ (x, t)
and
lim
β→0+
Ψ0 (Cβ) =0 A∞ (β)
for any x ∈ Ω , t ∈ [0, +∞), α > 0, and any given C > 0. Theorem 4.2. Suppose A satisfies condition (A∞ ). Let the Condition (F1) hold and the function Ψ in − assumption (F1) satisfies the assumptions (Ψ∞ ) and (Ψ∞ ). Then the problem (1.1) has a weak solution.
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Proof. By (F1) we have |F (x, t)| ≤ C1 Ψ (x, |t|) + h(x)|t|,
∀(x, t) ∈ Ω × R.
Then J is coercive because of the following inequality J (u) = B(u) − F(u) ≥ B(x, |∇u|) dx − |F (x, u)| dx Ω Ω ≥ b2 A(x, |∇u|) dx − C1 Ψ (x, |u|) dx − 2|h|Ψ ′ |u|Ψ Ω Ω |u| |∇u| dx − C1 dx − C|u|Ψ Ψ x, |u|Ψ = b2 A x, ∥u∥ ∥u∥ |u|Ψ Ω Ω |∇u| |u| dx − C1 Ψ∞ (|u|Ψ )|u|Ψ dx − C|u|Ψ ≥ b2 A∞ (∥u∥)∥u∥ A x, Ψ x, ∥u∥ |u|Ψ Ω Ω = b2 A∞ (∥u∥)∥u∥ − C1 Ψ∞ (|u|Ψ )|u|Ψ − C|u|Ψ ≥ b2 A∞ (∥u∥)∥u∥ − CΨ∞ (C∥u∥)∥u∥ − C∥u∥ = (b2 A∞ (∥u∥) − CΨ∞ (C∥u∥) − C)∥u∥ → +∞, as ∥u∥ → +∞. By Proposition 3.1 and Lemma 4.1, it is easy to verify that J is weakly lower semi-continuous. Then J has a minimum point u in X and u is a weak solution of (1.1), which completes the proof. Lemma 4.2. Suppose A satisfies condition (A∞ ). Then under Assumptions (F1) and (F2), the functional J satisfies the (P.S.) condition. Proof. Suppose that {un } ⊂ X, |J (un )| ≤ C0 for some C0 ∈ R, and J ′ (un ) → 0 in X ∗ as n → ∞. Let c := supn J (un ) and β ∈ ( θ1 , bb21 ) for large n. From Proposition 2.1, Theorem 2.10, (F2) and (A3 ), for big ∥un ∥ we have, c + 1 + ∥un ∥ ≥ J (un ) − β⟨J ′ (un ), un ⟩ 2 = B(x, |∇un |) dx − β b(x, |∇un |)|∇un | dx + βf (x, un )un − F (x, un ) dx Ω Ω Ω ≥ b2 A(x, |∇un |) dx − b1 β A(x, |∇un |) dx + βf (x, un )un − F (x, un ) dx Ω Ω Ω ≥ b2 − b1 β A(x, |∇un |) dx + βθ − 1 F (x, un ) dx Ω Ω ≥ b2 − b1 β A(x, |∇un |) dx Ω |∇un | = b2 − b1 β A x, ∥un ∥ dx ∥un ∥ Ω |∇un | ≥ b2 − b1 β A∞ (∥un ∥)∥un ∥ A x, dx ∥un ∥ Ω = b2 − b1 β A∞ (∥un ∥)∥un ∥. Since b2 − b1 β > 0 and A∞ (t) → +∞ as t → +∞, we obtain the boundedness of {un } in X by the above inequality. Without loss of generality, we assume un ⇀ u, then ⟨J ′ (un ) − J ′ (u), un − u⟩ → 0. Since J ′ is of type (S+ ), we have un → u in X.
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Theorem 4.3. Suppose A satisfies condition (A∞ ) and (A0 ). Let the function Ψ in (F1) satisfies (Ψ0 ) and h(x) ≡ 0 in (F1). Then under Conditions (F1)–(F3), the problem (1.1) has a nontrivial solution. Proof. We will show that the functional J satisfies the geometry condition of the Mountain Pass Theorem. By Lemma 4.2, A satisfies (P.S.) condition in X. By Assumptions (F1) and (F3), there exists some ϵ > 0 small enough such that |F (x, t)| ≤ ϵA(x, |t|) + C(ϵ)Ψ (x, |t|),
∀(x, t) ∈ Ω × [0, +∞).
So by (A0 ) and (Ψ0 ) we have for small ∥u∥ and ϵ small enough, J (u) ≥ b2 A(x, |∇u|) dx − ϵ A(x, |u|) dx − C(ϵ) Ψ (x, |u|) dx Ω Ω Ω |∇u| |u| |u| ≥ b2 A x, ∥u∥ dx − ϵ A x, |u|A dx − C(ϵ) Ψ x, |u|Ψ dx ∥u∥ |u|A |t|Ψ Ω Ω Ω |u| |∇u| dx − ϵA0 (|u|A )|u|A A x, dx ≥ b2 A∞ (∥u∥)∥u∥ A x, ∥u∥ |u|A Ω Ω |u| − C(ϵ)Ψ0 (|u|Ψ )|u|Ψ Ψ x, dx |u|Ψ Ω = b2 A∞ (∥u∥)∥u∥ − ϵA0 (|u|A )|u|A − C(ϵ)Ψ0 (|u|A∗ )|u|A∗ ≥ b2 A∞ (∥u∥)∥u∥ − ϵCA0 (C∥u∥)∥u∥ − CC(ϵ)Ψ0 (C∥u∥)∥u∥ b2 ≥ A∞ (∥u∥)∥u∥ − CC(ϵ)Ψ0 (C∥u∥)∥u∥. 2 Since limβ→0+
Ψ0 (Cβ) A∞ (β)
= 0, there exist r > 0 and δ > 0 such that J (u) ≥ δ > 0 for every ∥u∥ = r.
From the Assumption (F2), (B1 ) and (B2 ), there exist constants C1 , C2 > 0 such that b1
F (x, s) ≥ C1 |s|θ > C1 |s| b2 ≥ C2 A(x, |s|),
for |s| ≥ M, ∀x ∈ Ω .
For w ∈ X \ {0} and big t > 0, in view of the above inequality and θ > bb21 we have J (tw) = B(x, |t∇w|) dx − F (x, tw) dx Ω Ω ≤ b1 A(x, |t∇w|) dx − C1 tθ |w|θ dx − C Ω Ω b1 b1 θ b2 b2 |∇w| dx − C1 t |w|θ dx − C ≤ C3 t Ω
→ −∞,
Ω
as t → +∞.
Obviously we have J (0) = 0, so J satisfies the geometry conditions of the Mountain Pass Theorem in [11]. Then J admits at least one nontrivial critical point which corresponds to the weak solution of (1.1). In the following text, we will use the following assumption on f . (F4) f (x, −t) = −f (x, t), ∀t ∈ R, x ∈ Ω . Thanks to the Assumption (F4), the functional J is even. We can make an application of Fountain theorem and Dual Fountain theorem to get infinitely many solutions to (1.1). Theorem 4.4. Suppose A satisfies condition (A∞ ). Let (F1), (F2) and (F4) hold, the function Ψ in assumption (F1) satisfies assumption (Ψ∞ ) and h(x) ≡ 0 in (F1). Then (1.1) has a sequence of weak solutions {±uk }∞ k=1 such that J (±uk ) → +∞ as k → +∞.
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We will use Fountain theorem to prove Theorem 4.4. By Propositions 2.4 and 2.6, X is a reflexive and separable Banach space, there exist {ej } ⊂ X and {e∗j } ⊂ X ∗ such that X = span{ej : j = 1, 2, . . .},
X ∗ = span{e∗j : j = 1, 2, . . .}
in which ⟨ei , e∗j ⟩
1, = 0,
i = j, i ̸= j.
Denote Xj = span{ej }, Yk = ⊕kj=1 Xj , Zk = ⊕∞ j=k Xj . Lemma 4.3. If Ψ ≪ A∗ , denote Ψ (x, |u|) dx : ∥u∥ = 1, u ∈ Zk , βk = sup Ω
then limk→+∞ βk = 0. Proof. It is obvious that 0 < βk+1 < βk . Then there exists a β > 0 such that βk → β as k → +∞. By the definition of βk , for any k > 0, there exists uk ∈ Zk , such that ∥uk ∥ = 1 and 0 ≤ β − |uk |Ψ < k1 . Since ∥uk ∥ = 1, there exists a subsequence of {uk } (still denoted by {uk }) such that uk ⇀ u in W 1,A (Ω ), and by the definition of Zk we have ⟨e∗j , u⟩ = lim ⟨e∗j , u⟩ = 0, k→+∞
∀j = 1, 2, . . . .
Then u = 0, which implies that uk ⇀ 0 in W 1,A (Ω ). By Theorem 2.10 the embedding W 1,A (Ω ) ↩→↩→ LΨ (Ω ) is compact, then uk → 0 in LΨ (Ω ), which implies that βk → 0 by Proposition 2.3. The proof of the lemma is completed. Lemma 4.4 (Fountain Theorem, See [11]). Assume (M1) X is a Banach space, J ∈ C 1 (X, R) is an even functional. If for each k ∈ N, there exists ρk > rk > 0 such that (M2) inf u∈Zk ,∥u∥=rk J (u) → +∞ as k → +∞. (M3) maxu∈Yk ,∥u∥=ρk J (u) ≤ 0 (M4) J satisfies (P.S.)c condition for every c > 0. Then J admits a sequence of critical values tending to +∞. The proof of Theorem 4.4. By the assumption (F4), F is even, which implies J = B−F is also even. Further more, by Lemma 4.2, A satisfies the (P.S.)c condition. We need only to prove that there exists ρk > rk > 0 such that condition (M2) and (M3) in Lemma 4.4 hold. (M2). Let u ∈ Zk . By (F1), Proposition 2.3 and the definition of βk , we have J (u) = B(u) − F(u) ≥ B(x, |∇u|) dx − |F (x, u)| dx Ω
Ω
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1 A(x, |∇u|) dx − C Ψ (x, |u|) dx − 2|h|Ψ ′ |u|Ψ Ω Ω |u| |∇u| 1 dx − C dx − C|u|Ψ Ψ x, ∥u∥ = b2 A x, ∥u∥ ∥u∥ ∥u∥ Ω Ω |u| |∇u| 1 Ψ∞ (∥u∥)∥u∥ 2 ∥u∥ dx − C Ψ x, dx − C ≥ b2 A∞ (∥u∥)∥u∥ A x, ∥u∥ ∥u∥ Ω Ω ≥ b2
1 Ψ∞ (∥u∥)∥u∥βk − C 2 ∥u∥. ≥ b2 A∞ (∥u∥)∥u∥ − C
(4.1)
By Assumption (F2), (B1 ) and (B2 ), there exist constants C1 , C2 > 0 such that b1
F (x, s) ≥ C1 |s|θ > C1 |s| b2 ≥ C2 A(x, |s|),
for |s| ≥ M, ∀x ∈ Ω .
(4.2)
At the same time from assumption (F1), (A∞ ) and (Ψ∞ ), for |s| = tM we have CΨ∞ (t)tΨ (x, M ) + C ≥ CΨ (x, |s|) + C ≥ F (x, s), A(x, |s|) ≥ A∞ (t)tA(x, M ). Then assumption θ >
b1 b2 ,
(4.3)
(4.2) and (4.3) implies lim
t→+∞
A∞ (t) = 0. Ψ∞ (t)
Since βk → 0 as k → +∞, by the continuity of the function A∞ and Ψ∞ , for big k there exists rk (= rk (βk ) → +∞ as k → +∞) such that 1 1 βk Ψ (rk ) + C 2 . b2 A∞ (rk ) = C 2 Set ∥u∥ = rk in the inequality (4.1) we have 1 Ψ∞ (∥u∥)∥u∥βk − C 2 ∥u∥ J (u) ≥ b2 A∞ (∥u∥)∥u∥ − C 1 = b2 A∞ (rk )rk → +∞ as k → +∞. 2 (M3). By (4.2), for any w ∈ Yk with ∥w∥ = 1 and ρk = t > rk , we have J (tw) = B(x, |t∇w|) dx − F (x, tw) dx Ω Ω θ ≤ b1 A(x, |t∇w|) dx − C1 t |w|θ dx − C Ω Ω b1 b1 θ b2 b2 ≤ C3 t |∇w| dx − C1 t |w|θ dx − C Ω b1 b2
= C3 t ∥w∥
Ω b1 b2
b 1, 1 W b2
θ
(Ω)
− C1 t |w|θLθ (Ω) dx − C.
Since all norms in finite dimensional space Yk are equivalent and θ > The conclusion of Theorem 4.4 is obtained by Lemma 4.4.
p1 p2 ,
we have J (tw) → −∞.
Acknowledgments The authors are very grateful to the reviewers for their valued comments. This research is supported by the National Natural Science Foundation of China (NSFC 11471147) and Fundamental Research Funds for the Central Universities (lzujbky-2014-25).
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