Existence and multiplicity of positive solutions for a class of p ( x )-Kirchhoff type equations
© Ma et al; licensee Springer. 2012
Received: 24 September 2011
Accepted: 13 February 2012
Published: 13 February 2012
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 sub-supersolution 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.
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-: = infΩ p(x) ≤ p+ := supΩ p(x) < +∞, , M(t) is a function with and satisfies the following condition:
(M0) M(t): [0, +∞) → (m0, +∞) is a continuous and increasing function with m0 > 0.
The operator -div(|∇u|p(x)-2∇u) := -Δp(x)u is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; 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 [1–3]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian 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 . The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [7–11] for an overview of and references on this subject, and to [12–16] for the study of the variable exponent equations and the corresponding variational problems.
is related to the stationary analogue of the Equation (1.2). Equation (1.3) received much attention only after Lions  proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [19–22]. 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 [23–26]. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [27–29]) and p(x)-Laplacian (see [30–33]).
Many authors have studied the Neumann problems involving the p-Laplacian, see e.g., [34–36] 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 , Fan and Deng studied the Neumann problems with p(x)-Laplacian, with the nonlinear potential f(x, u) under appropriate assumptions. By using the sub-supersolution 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 [34–36] to the p(x)-Kirchhoff case. For simplicity we shall restrict to the 0-Neumann boundary value problems, but the methods used in this article are also suitable for the inhomogeneous Neumann boundary value problems.
The main results of this article are the following theorems. Throughout the article we always suppose that the condition (M0) holds.
Then , λ* ≥ 0 and (λ*, +∞) ⊂ Λ. Moreover, for every λ > λ* problem has a minimal positive solution uλ in [0,w1], where w1 is the unique solution of and if λ* < λ2 < λ1.
Then for each λ ∈ (λ*, +∞), has at least two positive solutions uλ and vλ, where uλ is a local minimizer of the energy functional and uλ ≤ vλ.
where M2, c3 and c4 are positive constants, and 1 ≤ r(x) < p(x) for . Then λ* = 0.
(2) If f satisfies (1.4)-(1.8), then λ* ∈ Λ.
So the conditions (M0) and (1.7) are satisfied.
The underlying idea for proving Theorems 1.1-1.3 is similar to the one of . 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 1.1-1.3, we use the results of  on the global C1,αregularity of the weak solutions for the p(x)-Laplacian equations. The main method used in this article is the sub-supersolution 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 p-Laplacian equations are very complicated (see e.g. [38–41]). 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 sub-supersolution method for the problem based on the regularity results. In Section 4, we give the proof of Theorems 1.1-1.3.
In order to discuss problem , we need some theories on W1,p(x)(Ω) which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces W1,p(x)(Ω) which will be used later (for details, see ). 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.
Denote by the closure of in W1,p(x)(Ω). The spaces Lp(x)(Ω), W1,p(x)(Ω) and are all separable Banach spaces. When p- > 1 these spaces are reflexive.
Then ||u||λ is a norm on W1,p(x)(Ω) equivalent to .
By the definition of ||u||λ we have the following
u k → u in measure in Ω and .
then there is a compact embedding W1,p(x)(Ω) ↪ Lq(x)(Ω).
where 〈·, ·〉 is the duality pairing between X and X*. Therefore, the p(x)-Kirchhoff-Laplace operator is the derivative operator of Φ in the weak sense. We have the following properties about the derivative operator of Φ.
Φ': X → X* is a continuous, bounded and strictly monotone operator;
Φ' is a mapping of type (S+), i.e., if u n ⇀ u in X and , then u n → u in X;
Φ'(u): X → X* is a homeomorphism;
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 Sub-supersolution principle
In this section we give a general principle of sub-supersolution method for the problem based on the regularity results and the comparison principle.
In this article, we need the global regularity results for the weak solution of . Applying Theorems 4.1 and 4.4 of  and Theorem 1.3 of , we can easily get the following results involving of the regularity of weak solutions of .
- (2)Let u ∈ X ∩ L∞ (Ω) be a solution of . If the function p is log-Hölder continuous on , i.e., there is a positive constant H such that(3.2)
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 (M0), applying Theorem 1.1 of , we have the following strong maximum principle.
where , M(t) ≥ m0 > 0, 0 ≤ d(x) ∈ L∞(Ω), with p(x) ≤ q(x) ≤ p* (x), then u > 0 in Ω.
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) = Kλ(h) = u the unique solution of (3.3λ). K = Kλ is called the solution operator for (3.3λ). From the regularity results and the embedding theorems we can obtain the properties of the solution operator K as follows.
If p is log-Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
If p is Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
Using the similar proof to , we have
Proposition 3.4. If and h ≥ 0, where satisfies (2.1), then K(h) ≥ 0. If p ∈ C1(Ω), h ∈ L∞(Ω) and h ≥ 0, then K(h) > 0 on .
Now we give a comparison principle as follows.
with φ ≥ 0 and u ≤ v on ∂ Ω, , then u ≤ v in Ω.
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 (M0), that is, K(u) ≤ K(v) if u ≤ v.
In this article we will use the following sub-supersolution 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., ) and is similar to that given in  for Dirichlet problems involving the p(x)-Laplacian.
then has a minimal solution u * and a maximal solution v* in the order interval [u0,v0], i.e., u0 ≤ u* ≤ v* ≤ v0 and if u is any solution of such that u0 ≤ u ≤ v0, then u* ≤ u ≤ v*.
The critical points of Jλ are just the solutions of . Many authors, for example, Chang , Brezis and Nirenberg  and Ambrosetti et al. , have combined the sub-supersolution 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 C1-topology is also a local minimizer in the H1-topology. Such lemma have been extended to the case of the p-Laplacian equations (see [43, 49]) and also to the case of the p(x)-Laplacian equations (see [12, Theorem 3.1]). In , Fan extended the Brezis-Nirenberg type theorem to the case of the p(x)-Kirchhoff [50, Theorem 1.1]. The Theorem 1.1 of  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 Jλ in the -topology, then u is also a local minimizer of Jλ in the X-topology.
4 Proof of theorems
The proof of Theorem 1.1 consists of the following several Lemmata 4.1-4.6.
Lemma 4.1. Let (1.4) hold. Then λ > 0 if λ ∈ Λ.
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 .
This shows that w1 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 λ1 ∈ Λ and .
Lemma 4.3. Let (1.4) and (1.5) hold. If λ0 ∈ Λ, then λ ∈ Λ for all λ > λ0.
thanks to (M0). This shows that is a supersolution of . We know that 0 is a subsolution of By Theorem 3.3, has a solution uλ such that . By Proposition 3.4, uλ > 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 uλ of such that if λ* < λ2 < λ1.
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 λ1, λ2 ∈ Λ and λ2 < λ < λ1. Suppose that and are the positive solutions of and respectively and . Then there exists a positive solution vλ of such that and vλ is a global minimizer of the restriction of Jλ to the order interval .
and λ2 < λ < λ1, since K is increasing operator, we obtain that . So , and vλ is a positive solution of . It is easy to see that there exists a constant c such that for . Hence vλ 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 λ1, λ2 ∈ Λ and λ2 < λ1. Suppose that and are the positive solutions of and respectively. Then on .
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 λ1, λ2 ∈ Λ such that λ2 < λ < λ1 and let be as in Lemma 4.5.
We claim that uλ is a local minimizer of Jλ in the X-topology.
Indeed, Lemma 4.6 implies that on . It follows that there is a C0-neighborhood U of uλ such that , consequently uλ is a local minimizer of Jλ in the C0-topology, and of course, also in the C1-topology. By Theorem 3.4, uλ is also a local minimizer of Jλ in the X-topology.
and denote by the energy functional corresponding to (4.4λ). By the definition of , we have for every u ∈ X. Hence, for each solution u of (4.4λ), we have that u ≥ uλ, consequently and u is also a solution of . It is easy to see that and are a subsolution and a supersolution of (4.4λ) respectively. By Theorems 3.3 and 1.2, there exists such that is a solution of (4.4λ) and is a local minimizer of in the C1-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 uλ is a local minimizer of in the C1-topology, and so also in the X-topology. We can assume that uλ is a strictly local minimizer of in the X-topology, 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., ). It follows from the condition (1.7) and (1.8) that (see e.g., ). Using the mountain pass lemma (see ), we know that (4.4λ) has a solution vλ such that vλ ≠ uλ. vλ, as a solution of (4.4λ), must satisfy vλ ≥ uλ, and vλ is also a solution of . The proof of Theorem 1.2 is complete.
where c5 is a positive constant. This shows that Jλ(u) → +∞ as ||u||λ → +∞, that is, Jλ is coercive. In view of Proposition 2.5. (iv), the condition (1.10) also implies that Jλ is weakly sequentially lower semi-continuous. Thus Jλ has a global minimizer u0. Put v0(x) = |u0(x)| for . It is easy to see that Jλ(v0) ≤ Jλ(u0), consequently, v0 is a global minimizer of Jλ 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 uλ such that Jλ(uλ) ≤ 0.
Proof. Let λ > λ*. Take λ2 ∈ (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 Jλ(uλ) ≤ Jλ(0) = 0.
which shows that u* is a solution of . Obviously u* ≥ 0 and . Hence u* is a positive solution of and λ* ∈ Λ.
The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030 and 10971087), NWNU-LKQN-10-21.
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