Abstract
In this article, we study the existence and multiplicity of positive solutions for the Neumann boundary value problems involving the p(x)Kirchhoff of the form
Using the subsupersolution method and the variational method, under appropriate assumptions on f and M, we prove that there exists λ_{* }> 0 such that the problem has at least two positive solutions if λ > λ_{*}, at least one positive solution if λ = λ_{* }and no positive solution if λ < λ_{*}. To prove these results we establish a special strong comparison principle for the Neumann problem.
2000 Mathematical Subject Classification: 35D05; 35D10; 35J60.
Keywords:
p(x)Kirchhoff; positive solution; subsupersolution method; comparison principle1 Introduction
In this article we study the following problem
where Ω is a bounded domain of ℝ^{N }with smooth boundary ∂Ω and N ≥ 1, is the outer unit normal derivative, λ ∈ ℝ is a parameter, with 1 < p^{}: = inf_{Ω }p(x) ≤ p^{+ }:= sup_{Ω }p(x) < +∞, , M(t) is a function with and satisfies the following condition:
(M_{0}) M(t): [0, +∞) → (m_{0}, +∞) is a continuous and increasing function with m_{0 }> 0.
The operator div(∇u^{p(x)2}∇u) := Δ_{p(x)}u is said to be the p(x)Laplacian, and becomes pLaplacian when p(x) ≡ p (a constant). The p(x)Laplacian possesses more complicated nonlinearities than the pLaplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [13]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermorheological viscous flows of nonNewtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [4,5]. Another field of application of equations with variable exponent growth conditions is image processing [6]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [711] for an overview of and references on this subject, and to [1216] for the study of the variable exponent equations and the corresponding variational problems.
The problem is a generalization of the stationary problem of a model introduced by Kirchhoff [17]. More precisely, Kirchhoff proposed a model given by the equation
where ρ, ρ_{0}, h, E, L are constants, which extends the classical D'Alembert's wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinguishing feature of Equation (1.2) is that the equation contains a nonlocal coefficient which depends on the average , and hence the equation is no longer a pointwise identity. The equation
is related to the stationary analogue of the Equation (1.2). Equation (1.3) received much attention only after Lions [18] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [1922]. Moreover, nonlocal boundary value problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density [2326]. The study of Kirchhoff type equations has already been extended to the case involving the pLaplacian (for details, see [2729]) and p(x)Laplacian (see [3033]).
Many authors have studied the Neumann problems involving the pLaplacian, see e.g., [3436] and the references therein. In [34,35] the authors have studied the problem in the cases of p(x) ≡ p = 2, M(t) ≡ 1 and of p(x) ≡ p > 1, M(t) ≡ 1, respectively. In [36], Fan and Deng studied the Neumann problems with p(x)Laplacian, with the nonlinear potential f(x, u) under appropriate assumptions. By using the subsupersolution method and variation method, the authors get the multiplicity of positive solutions of with M(t) ≡ 1. The aim of the present paper is to generalize the main results of [3436] to the p(x)Kirchhoff case. For simplicity we shall restrict to the 0Neumann boundary value problems, but the methods used in this article are also suitable for the inhomogeneous Neumann boundary value problems.
In this article we use the following notations:
Λ = {λ ∈ ℝ: there exists at least a positive solution of },
The main results of this article are the following theorems. Throughout the article we always suppose that the condition (M_{0}) holds.
Theorem 1.1. Suppose that f satisfies the following conditions:
and
Then , λ_{* }≥ 0 and (λ_{*}, +∞) ⊂ Λ. Moreover, for every λ > λ_{* }problem has a minimal positive solution u_{λ }in [0,w_{1}], where w_{1 }is the unique solution of and if λ_{* }< λ_{2 }< λ_{1}.
Theorem 1.2. Under the assumptions of Theorem 1.1, also suppose that there exist positive constants M, c_{1 }and c_{2 }such that
where and 1 ≤ q(x) < p*(x) for , μ ∈ (0,1) such that
where and M_{1 }> 0, such that
Then for each λ ∈ (λ_{*}, +∞), has at least two positive solutions u_{λ }and v_{λ}, where u_{λ }is a local minimizer of the energy functional and u_{λ }≤ v_{λ}.
Theorem 1.3. (1) Suppose that f satisfies (1.4),
and the following conditions:
where M_{2}, c_{3 }and c_{4 }are positive constants, and 1 ≤ r(x) < p(x) for . Then λ_{* }= 0.
(2) If f satisfies (1.4)(1.8), then λ_{*} ∈ Λ.
Example 1.1. Let M(t) = a + bt, where a and b are positive constants. It is clear that
So the conditions (M_{0}) and (1.7) are satisfied.
The underlying idea for proving Theorems 1.11.3 is similar to the one of [36]. The special features of this class of problems considered in the present article are that they involve the nonlocal coefficient M(t). To prove Theorems 1.11.3, we use the results of [37] on the global C^{1,α }regularity of the weak solutions for the p(x)Laplacian equations. The main method used in this article is the subsupersolution method for the Neumann problems involving the p(x)Kirchhoff. A main difficulty for proving Theorem 1.1 is that a special strong comparison principle is required. It is well known that, when p ≠ 2, the strong comparison principles for the pLaplacian equations are very complicated (see e.g. [3841]). In [13,42,43] the required strong comparison principles for the Dirichlet problems have be established, however, they cannot be applied to the Neumann problems. To prove Theorem 1.1, we establish a special strong comparison principle for the Neumann problem (see Lemma 4.6 in Section 4), which is also valid for the inhomogeneous Neumann boundary value problems.
In Section 2, we give some preliminary knowledge. In Section 3, we establish a general principle of subsupersolution method for the problem based on the regularity results. In Section 4, we give the proof of Theorems 1.11.3.
2 Preliminaries
In order to discuss problem , we need some theories on W^{1,p(x) }(Ω) which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces W^{1,p(x) }(Ω) which will be used later (for details, see [17]). Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere.
Write
and
with the norm
and
with the norm
Denote by the closure of in W^{1,p(x) }(Ω). The spaces L^{p(x) }(Ω), W^{1,p(x) }(Ω) and are all separable Banach spaces. When p^{ }> 1 these spaces are reflexive.
Let λ > 0. Define for u ∈ W^{1,p(x) }(Ω),
Then u_{λ }is a norm on W^{1,p(x) }(Ω) equivalent to .
By the definition of u_{λ }we have the following
Proposition 2.1. [11,14]Put for λ > 0 and u ∈ W^{1,p(x) }(Ω). We have:
Proposition 2.2. [14]If u, u_{k }∈ W^{1,p(x) }(Ω), k = 1,2,..., then the following statements are equivalent each other:
(iii) u_{k }→ u in measure in Ω and .
Proposition 2.3. [14]Let . If satisfies the condition
then there is a compact embedding W^{1,p(x) }(Ω) ↪ L^{q(x) }(Ω).
Proposition 2.4. [14]The conjugate space of L^{p(x) }(Ω) is L^{q(x) }(Ω), where . For any u ∈ L^{p(x) }(Ω) and v ∈ L^{q(x) }(Ω), we have the following Höldertype inequality
Now, we discuss the properties of p(x)KirchhoffLaplace operator
where λ > 0 is a parameter. Denotes
For simplicity we write X = W^{1,p(x) }(Ω), denote by u_{n }⇀ u and u_{n }→ u the weak convergence and strong convergence of sequence {u_{n}} in X, respectively. It is obvious that the functional Φ is a Gâteaux differentiable whose Gâteaux derivative at the point u ∈ X is the functional Φ'(u) ∈ X*, given by
where 〈·, ·〉 is the duality pairing between X and X*. Therefore, the p(x)KirchhoffLaplace operator is the derivative operator of Φ in the weak sense. We have the following properties about the derivative operator of Φ.
Proposition 2.5. If (M_{0}) holds, then
(i) Φ': X → X* is a continuous, bounded and strictly monotone operator;
(ii) Φ' is a mapping of type (S_{+}), i.e., if u_{n }⇀ u in X and , then u_{n }→ u in X;
(iii) Φ'(u): X → X* is a homeomorphism;
(iv) Φ is weakly lower semicontinuous.
Proof. Applying the similar method to prove [15, Theorem 2.1], with obvious changes, we can obtain the conclusions of this proposition.
3 Subsupersolution principle
In this section we give a general principle of subsupersolution method for the problem based on the regularity results and the comparison principle.
Definition 3.1. u ∈ X is called a weak solution of the problem if for all v ∈ X,
In this article, we need the global regularity results for the weak solution of . Applying Theorems 4.1 and 4.4 of [44] and Theorem 1.3 of [37], we can easily get the following results involving of the regularity of weak solutions of .
Proposition 3.1. (1) If f satisfies (1.6), then u ∈ L^{∞}(Ω) for every weak solution u of .
(2) Let u ∈ X ∩ L^{∞ }(Ω) be a solution of . If the function p is logHölder continuous on , i.e., there is a positive constant H such that
(3) If in (2), the condition (3.2) is replaced by that p is Hölder continuous on , then for some α ∈ (0,1).
For u, v ∈ S(Ω), we write u ≤ v if u(x) ≤ v(x) for a.e. x ∈ Ω. In view of (M_{0}), applying Theorem 1.1 of [16], we have the following strong maximum principle.
Proposition 3.2. Suppose that , u ∈ X, u ≥ 0 and in Ω. If
where , M(t) ≥ m_{0 }> 0, 0 ≤ d(x) ∈ L^{∞}(Ω), with p(x) ≤ q(x) ≤ p* (x), then u > 0 in Ω.
Definition 3.2. u ∈ X is called a subsolution (resp. supersolution) of if for all v ∈ X with v ≥ 0, u ≤ 0 (resp. ≥) on ∂Ω and
Theorem 3.1. Let λ > 0 and satisfies (2.1). Then for each , the problem
has a unique solution u ∈ X.
Proof. According to Propositions 2.3 and 2.4, (for any v ∈ X) defines a continuous linear functional on X. Since Φ' is a homeomorphism, has a unique solution.
Let satisfy (2.1). For , we denote by K(h) = K_{λ}(h) = u the unique solution of (3.3_{λ}). K = K_{λ }is called the solution operator for (3.3_{λ}). From the regularity results and the embedding theorems we can obtain the properties of the solution operator K as follows.
Proposition 3.3. (1) The mapping is continuous and bounded. Moreover, the mapping is completely continuous since the embedding X ↪ L^{q(x) }(Ω) is compact.
(2) If p is logHölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
(3) If p is Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
Using the similar proof to [36], we have
Proposition 3.4. If and h ≥ 0, where satisfies (2.1), then K(h) ≥ 0. If p ∈ C^{1}(Ω), h ∈ L^{∞}(Ω) and h ≥ 0, then K(h) > 0 on .
Now we give a comparison principle as follows.
Theorem 3.2. Let u, v ∈ X, . If
with φ ≥ 0 and u ≤ v on ∂Ω, , then u ≤ v in Ω.
Proof. Taking φ = (u  v)^{+ }as a test function in (3.4), we have
Using the similar proof to Theorem 2.1 of [15] with obvious changes, we can show that
Therefore, we get 〈Φ'(u)  Φ'(v), φ〉 = 0. Proposition 2.5 implies that φ ≡ 0 or u ≡ v in Ω. It follows that u ≤ v in Ω.
It follows from Theorem 3.2 that the solution operator K is increasing under the condition (M_{0}), that is, K(u) ≤ K(v) if u ≤ v.
In this article we will use the following subsupersolution principle, the proof of which is based on the well known fixed point theorem for the increasing operator on the order interval (see e.g., [45]) and is similar to that given in [12] for Dirichlet problems involving the p(x)Laplacian.
Theorem 3.3. (A subsupersolution principle) Suppose that u_{0}, v^{0 }∈ X ∩ L^{∞}(Ω), u_{0 }and v^{0 }are a subsolution and a supersolution of respectively, and u_{0 }≤ v^{0}. If f satisfies the condition:
then has a minimal solution u_{* }and a maximal solution v* in the order interval [u_{0},v^{0}], i.e., u_{0 }≤ u_{* }≤ v* ≤ v^{0 }and if u is any solution of such that u_{0 }≤ u ≤ v^{0}, then u_{* }≤ u ≤ v*.
The energy functional corresponding to is
The critical points of J_{λ }are just the solutions of . Many authors, for example, Chang [46], Brezis and Nirenberg [47] and Ambrosetti et al. [48], have combined the subsupersolution method with the variational method and studied successfully the semilinear elliptic problems, where a key lemma is that a local minimizer of the associated energy functional in the C^{1}topology is also a local minimizer in the H^{1}topology. Such lemma have been extended to the case of the pLaplacian equations (see [43,49]) and also to the case of the p(x)Laplacian equations (see [12, Theorem 3.1]). In [50], Fan extended the BrezisNirenberg type theorem to the case of the p(x)Kirchhoff [50, Theorem 1.1]. The Theorem 1.1 of [50] concerns with the Dirichlet problems, but the method for proving the theorem is also valid for the Neumann problems. Thus we have the following
Theorem 3.4. Let λ > 0 and (1.6) holds. If is a local minimizer of J_{λ }in the topology, then u is also a local minimizer of J_{λ }in the Xtopology.
4 Proof of theorems
In this section we shall prove Theorems 1.11.3. Since only the positive solutions are considered, without loss of generality, we can assume that
otherwise we may replace f(x,t) by f^{(+)}(x,t), where
The proof of Theorem 1.1 consists of the following several Lemmata 4.14.6.
Lemma 4.1. Let (1.4) hold. Then λ > 0 if λ ∈ Λ.
Proof. Let λ ∈ Λ and u be a positive solution of . Taking v ≡ 1 as a test function in Definition 3.1. (1) yields
which implies λ > 0 because the value of the right side in (4.1) is positive.
Lemma 4.2. Let (1.4) and (1.5) hold. Then .
Proof. By Theorem 3.1, Propositions 3.4 and 3.3. (3), the problem
has a unique positive solution and w_{1}(x) ≥ ε > 0 for . We can assume ε ≤ 1. Put and λ_{1 }= 1 + M_{3}. Then
This shows that w_{1 }is a supersolution of the problem . Obviously 0 is a subsolution of . By Theorem 3.3, has a solution such that . By Proposition 3.4, on . So λ_{1 }∈ Λ and .
Lemma 4.3. Let (1.4) and (1.5) hold. If λ_{0 }∈ Λ, then λ ∈ Λ for all λ > λ_{0}.
Proof. Let λ_{0 }∈ Λ and λ > λ_{0}. Let be a positive solution of . Then, we have
thanks to (M_{0}). This shows that is a supersolution of . We know that 0 is a subsolution of By Theorem 3.3, has a solution u_{λ }such that . By Proposition 3.4, u_{λ }> 0 on . Thus λ ∈ Λ.
Lemma 4.4. Let (1.4) and (1.5) hold. Then for every λ > λ_{*}, there exists a minimal positive solution u_{λ }of such that if λ_{* }< λ_{2 }< λ_{1}.
Proof. The proof is similar to [36, Lemma 3.4], we omit it here.
Lemma 4.5. Let (1.4) and (1.5) hold. Let λ_{1}, λ_{2 }∈ Λ and λ_{2 }< λ < λ_{1}. Suppose that and are the positive solutions of and respectively and . Then there exists a positive solution v_{λ }of such that and v_{λ }is a global minimizer of the restriction of J_{λ }to the order interval .
It is easy to see that the global minimum of on X is achieved at some v_{λ }∈ X. Thus v_{λ }is a solution of the following problem
and λ_{2 }< λ < λ_{1}, since K is increasing operator, we obtain that . So , and v_{λ }is a positive solution of . It is easy to see that there exists a constant c such that for . Hence v_{λ }is a global minimizer of .
A key lemma of this paper is the following strong comparison principle.
Lemma 4.6 (A strong comparison principle). Let (1.4) and (1.5) hold. Let λ_{1}, λ_{2 }∈ Λ and λ_{2 }< λ_{1}. Suppose that and are the positive solutions of and respectively. Then on .
Proof. Since and on , in view of Lemma 4.4, there exist two positive constants b_{1 }≤ 1 and b_{2 }such that
Taking an ε > 0 sufficiently small such that
and
then
consequently, v_{ε }is a solution of the problem
where . With other words, , where is the solution operator of . Since , where , noting that is increasing, we have , that is, on .
The proof of Theorem 1.1 is complete. Let us now turn to the proof of Theorem 1.2.
Proof of Theorem 1.2. Let (1.4)(1.8) hold. Let λ > λ_{*}. Take λ_{1}, λ_{2 }∈ Λ such that λ_{2 }< λ < λ_{1 }and let be as in Lemma 4.5.
We claim that u_{λ }is a local minimizer of J_{λ }in the Xtopology.
Indeed, Lemma 4.6 implies that on . It follows that there is a C^{0}neighborhood U of u_{λ }such that , consequently u_{λ }is a local minimizer of J_{λ }in the C^{0}topology, and of course, also in the C^{1}topology. By Theorem 3.4, u_{λ }is also a local minimizer of J_{λ }in the Xtopology.
Define
and denote by the energy functional corresponding to (4.4_{λ}). By the definition of , we have for every u ∈ X. Hence, for each solution u of (4.4_{λ}), we have that u ≥ u_{λ}, consequently and u is also a solution of . It is easy to see that and are a subsolution and a supersolution of (4.4_{λ}) respectively. By Theorems 3.3 and 1.2, there exists such that is a solution of (4.4_{λ}) and is a local minimizer of in the C^{1}topology. As was noted above, we know that and is also a solution of . If , then the assertion of Theorem 1.2 already holds, hence we can assume that . Now u_{λ }is a local minimizer of in the C^{1}topology, and so also in the Xtopology. We can assume that u_{λ }is a strictly local minimizer of in the Xtopology, otherwise we have obtained the assertion of Theorem 1.2. It is easy to verify that, under the assumptions of Theorem 1.3, and satisfies the (P.S.) condition (see e.g., [30]). It follows from the condition (1.7) and (1.8) that (see e.g., [30]). Using the mountain pass lemma (see [51]), we know that (4.4_{λ}) has a solution v_{λ }such that v_{λ }≠ u_{λ}. v_{λ}, as a solution of (4.4_{λ}), must satisfy v_{λ }≥ u_{λ}, and v_{λ }is also a solution of . The proof of Theorem 1.2 is complete.
Proof of Theorem 1.3. (1) Let f satisfy (1.4), (1.9), and (1.10). For given any λ > 0, consider the energy functional J_{λ }defined by (3.3). By (1.10) and noting that r(x) < p(x) for , there is a positive constant M_{4 }such that
For u ∈ X with u_{λ }≥ 1, we have that
where c_{5 }is a positive constant. This shows that J_{λ}(u) → +∞ as u_{λ }→ +∞, that is, J_{λ }is coercive. In view of Proposition 2.5. (iv), the condition (1.10) also implies that J_{λ }is weakly sequentially lower semicontinuous. Thus J_{λ }has a global minimizer u_{0}. Put v_{0}(x) = u_{0}(x) for . It is easy to see that J_{λ}(v_{0}) ≤ J_{λ}(u_{0}), consequently, v_{0 }is a global minimizer of J_{λ }and is a positive solution of . This shows that λ ∈ Λ for all λ > 0. Hence λ_{* }= 0 and the statement (1) is proved.
To prove Theorem 1.3. (2) we give the following lemma.
Lemma 4.7. Let (1.4) and (1.5) hold. Then for each λ > λ_{*}, has a positive solution u_{λ }such that J_{λ}(u_{λ}) ≤ 0.
Proof. Let λ > λ_{*}. Take λ_{2 }∈ (λ_{*}, λ) and let be a positive solution of . then is a supersolution of . We know that 0 is a subsolution of . Analogous to the proof of Lemma 4.5, we can prove that has a positive solution such that . So J_{λ}(u_{λ}) ≤ J_{λ}(0) = 0.
Proof of Theorem 1.3. (2). Let (1.4)(1.8) hold. Let λ_{n }> λ_{* }and λ_{n }→ λ_{* }as n → +∞. By Lemma 4.7, for each n, has a positive solution such that , that is
Since is a solution of , we have that
It follows from (1.8) that there exists a positive constant c_{6 }such that
Thus, using condition (1.7), we have that
and consequently,
where the positive constant c_{7 }is independent of n. This shows that is bounded. Noting that λ_{n }→ λ_{* }> 0, we have that is bounded. Without loss of generality, we can assume that in X and for a.e. x ∈ Ω. By (1.6) and the L^{∞}(Ω)regularity results of [44], the boundedness of implies the boundedness of . By the regularity results of [37], the boundedness of implies the boundedness of , where α ∈ (0, 1) is a constant. Thus we have in . For every v ∈ X, since is a solution of , we have that, for each n,
Passing the limit of above equality as n → +∞, yields
which shows that u_{* }is a solution of . Obviously u_{* }≥ 0 and . Hence u_{* }is a positive solution of and λ_{* }∈ Λ.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030 and 10971087), NWNULKQN1021.
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