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,
(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
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
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 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
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
where
Then for each λ ∈ (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,
(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
Taking
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
In Section 2, we give some preliminary knowledge. In Section 3, we establish a general
principle of subsupersolution method for the problem
2 Preliminaries
In order to discuss problem
Write
and
with the norm
and
with the norm
Denote by
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
(1)
(2)
(3)
(4)
Proposition 2.2. [14]If u, u_{k }∈ W^{1,p(x) }(Ω), k = 1,2,..., then the following statements are equivalent each other:
(i)
(ii)
(iii) u_{k }→ u in measure in Ω and
Proposition 2.3. [14]Let
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
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
(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
Definition 3.1. u ∈ X is called a weak solution of the problem
In this article, we need the global regularity results for the weak solution 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
then
(3) If in (2), the condition (3.2) is replaced by that p is Hölder continuous on
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
where
Definition 3.2. u ∈ X is called a subsolution (resp. supersolution) of
Theorem 3.1. Let λ > 0 and
has a unique solution u ∈ X.
Proof. According to Propositions 2.3 and 2.4,
Let
Proposition 3.3. (1) The mapping
(2) If p is logHölder continuous on
(3) If p is Hölder continuous on
Using the similar proof to [36], we have
Proposition 3.4. If
Now we give a comparison principle as follows.
Theorem 3.2. Let u, v ∈ X,
with φ ≥ 0 and u ≤ v on ∂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
then
The energy functional corresponding to
The critical points of J_{λ }are just the solutions of
Theorem 3.4. Let λ > 0 and (1.6) holds. If
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
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
This shows that w_{1 }is a supersolution of the problem
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
thanks to (M_{0}). This shows that
Lemma 4.4. Let (1.4) and (1.5) hold. Then for every λ > λ_{*}, there exists a minimal positive solution u_{λ }of
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
Proof. Define
Define
It is easy to see that the global minimum of
and
and λ_{2 }< λ < λ_{1}, since K is increasing operator, we obtain that
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
Proof. Since
For
Taking an ε > 0 sufficiently small such that
and
then
consequently, v_{ε }is a solution of the problem
where
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
We claim that u_{λ }is a local minimizer of J_{λ }in the Xtopology.
Indeed, Lemma 4.6 implies that
Define
and
and denote by
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
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
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_{*},
Proof. Let λ > λ_{*}. Take λ_{2 }∈ (λ_{*}, λ) and let
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,
Since
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
Passing the limit of above equality as n → +∞, yields
which shows that u_{* }is a solution of
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.
References

Růžička, M: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

Mihăilescu, M, Rădulescu, V: A mulyiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc R Soc A. 462, 2625–2641 (2006). Publisher Full Text

Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv. 9, 33–66 (1987)

Antontsev, SN, Shmarev, SI: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal TMA. 60, 515–545 (2005)

Antontsev, SN, Rodrigues, JF: On stationary thermorheological viscous flows. Ann Univ Ferrara Sez Sci Mat. 52, 19–36 (2006). Publisher Full Text

Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 66(4), 1383–1406 (2006). Publisher Full Text