This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence and multiplicity of positive solutions for a class of p(x)-Kirchho type equations Boundary Value Problems 2012, 2012:16 doi:10.1186/1687-2770-2012-16 Ruyun Ma (mary@nwnu.edu.cn) Guowei Dai (daiguowei2009@126.com) Chenghua Gao (gaokuguo@163.com) ISSN 1687-2770 Article type Research Submission date 24 September 2011 Acceptance date 13 February 2012 Publication date 13 February 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/16 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Boundary Value Problems © 2012 Ma et al. ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Existence and multiplicity of positive solutions for a class of p(x)-Kirchhoff type equations Ruyun Ma, Guowei Dai ∗ and Chenghua Gao Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China ∗ Corresponding author: daiguowei@nwnu.edu.cn Email address: RM: mary@nwnu.edu.cn CG: gaokuguo@163.com 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        −M   Ω 1 p(x)  |∇u| p(x) + λ|u| p(x)  dx   div  |∇u| p(x)−2 ∇u  − λ|u| p(x)−2 u  = f(x, u) in Ω, ∂u ∂υ = 0 on ∂Ω. Using the sub-supersolution metho d and the variational method, under appropriate assumptions on f and M, we prove that there exists λ ∗ > 0 such that the problem 1 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. Keywords: p(x)-Kirchhoff; positive solution; sub-supersolution method; comparison principle. 2000 Mathematical Subject Classification: 35D05; 35D10; 35J60. 1 Introduction In this article we study the following problem        −M (t)  div  |∇u| p(x)−2 ∇u  − λ|u| p(x)−2 u  = f(x, u) in Ω, ∂u ∂υ = 0 on ∂Ω, (P f λ ) where Ω is a bounded domain of R N with smooth boundary ∂Ω and N ≥ 1, ∂u ∂υ is the outer unit normal derivative, λ ∈ R is a parameter, p = p(x) ∈ C 1 (Ω) with 1 < p − := inf Ω p(x) ≤ p + := sup Ω p(x) < +∞, f ∈ C(Ω × R, R), M(t) is a function with t :=  Ω 1 p(x)  |∇u| p(x) + λ|u| p(x)  dx 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) := −∆ p(x) u is said to be the p(x)-Laplacian, and be- comes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more compli- cated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of 2 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 character- ized 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 [6]. 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. The problem  P f λ 1  is a generalization of the stationary problem of a model introduced by Kirchhoff [17]. More precisely, Kirchhoff proposed a model given by the equation ρ ∂ 2 u ∂t 2 −   ρ 0 h + E 2L L  0     ∂u ∂x     2 dx   ∂ 2 u ∂x 2 = 0, (1.2) 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 co efficient ρ 0 h + E 2L  L 0   ∂u ∂x   2 dx which depends on the average 1 L  L 0   ∂u ∂x   2 dx, and hence the equation is 3 no longer a pointwise identity. The equation        −  a + b  Ω |∇u| 2 dx  ∆u = f(x, u) in Ω, u = 0 on ∂Ω (1.3) 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 im- portant 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 pop- ulation 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  P f λ 1  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 sub-supersolution method and variation method, the authors get the multiplicity of positive solutions of  P f λ 1  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. 4 In this article we use the following notations: F (x, t) = t  0 f(x, s) ds, Λ =  λ ∈ R : there exists at least a positive solution of  P f λ  , λ ∗ = inf Λ. 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: f(x, t) ≥ 0, f(x, t) ≡ 0 ∀x ∈ Ω, ∀t ≥ 0 (1.4) and for each x ∈ Ω, f(x, t) is nondecreasing with respect to t ≥ 0. (1.5) Then Λ = ∅, λ ∗ ≥ 0 and (λ ∗ , +∞) ⊂ Λ. Moreover, for every λ > λ ∗ problem  P f λ  has a minimal positive solution u λ in [0, w 1 ], where w 1 is the unique solution of (P 0 λ ) and u λ 1 < u λ 2 if λ ∗ < λ 2 < λ 1 . 5 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 f(x, t) ≤ c 1 + c 2 t q(x)−1 , ∀x ∈ Ω, ∀t ≥ M, (1.6) where q ∈ C(Ω) and 1 ≤ q(x) < p ∗ (x) for x ∈ Ω, µ ∈ (0, 1) such that  M(t) ≥ (1 − µ)M(t)t, (1.7) where  M(t) =  t 0 M(τ ) dτ and M 1 > 0, θ > p + 1−µ such that 0 < θF(x, t) ≤ tf(x, t), ∀x ∈ Ω, ∀t ≥ M 1 . (1.8) Then for each λ ∈ (λ ∗ , +∞),  P f 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), f(x, 0) ≤ f(x, t) for t > 0 and x ∈ Ω (1.9) and the following conditions: f(x, t) ≤ c 3 + c 4 t r(x)−1 , ∀x ∈ Ω, ∀t ≥ M 2 , (1.10) where M 2 , c 3 and c 4 are positive constants, r ∈ C(Ω) and 1 ≤ r(x) < p(x) for x ∈ Ω. Then λ ∗ = 0. 6 (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 M(t) ≥ a > 0. Taking µ = 1 2 , we have  M(t) = t  0 M(s) ds = at + 1 2 bt 2 ≥ 1 2 (a + bt)t = (1 − µ)M(t)t. So the conditions (M 0 ) and (1.7) are satisfied. The underlying idea for proving Theorems 1.1–1.3 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 1.1–1.3, 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 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  P f λ  (see Lemma 4.6 in Section 4), which is also valid for the inhomogeneous Neumann boundary value problems. 7 In Section 2, we give some preliminary knowledge. In Section 3, we establish a general principle of sub-supersolution method for the problem  P f λ  based on the regularity results. In Section 4, we give the proof of Theorems 1.1–1.3. 2 Preliminaries In order to discuss problem  P f λ  , 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 C + (Ω) =  h : h ∈ C(Ω), h(x) > 1 for any x ∈ Ω  and L p(x) (Ω) =    u ∈ S(Ω) :  Ω |u(x)| p(x) dx < +∞    with the norm |u| L p(x) (Ω) = |u| p(x) = inf    λ > 0 :  Ω     u(x) λ     p(x) dx ≤ 1    , and W 1,p(x) (Ω) =  u ∈ L p(x) (Ω) : |∇u| ∈ L p(x) (Ω)  with the norm u = u W 1,p(x) (Ω) = |u| L p(x) (Ω) + |∇u| L p(x) (Ω) . 8 Denote by W 1,p(x) 0 (Ω) the closure of C ∞ 0 (Ω) in W 1,p(x) (Ω) . The spaces L p(x) (Ω) , W 1,p(x) (Ω) and W 1,p(x) 0 (Ω) are all separable Banach spaces. When p − > 1 these spaces are reflexive. Let λ > 0. Define for u ∈ W 1,p(x) (Ω) , u λ = inf    σ > 0 :  Ω      ∇u σ     p(x) + λ    u σ    p(x)  dx ≤ 1    . Then u λ is a norm on W 1,p(x) (Ω) equivalent to u W 1,p(x) (Ω) . By the definition of u λ we have the following Proposition 2.1. [11, 14] Put ρ λ (u) =  Ω  |∇u| p(x) + λ |u| p(x)  dx for λ > 0 and u ∈ W 1,p(x) (Ω) . We have: (1) u λ ≥ 1 =⇒ u p − λ ≤ ρ λ (u) ≤ u p + λ ; (2) u λ ≤ 1 =⇒ u p + λ ≤ ρ λ (u) ≤ u p − λ ; (3) lim k→+∞ u k  λ = 0 ⇐⇒ lim k→+∞ ρ λ (u k ) = 0 (as k → +∞); (4) lim k→+∞ u k  λ = +∞ ⇐⇒ lim k→+∞ ρ λ (u k ) = +∞ (as k → +∞). 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) lim k→+∞ u k − u λ = 0; (ii) lim k→+∞ ρ λ (u k − u) = 0; (iii) u k → u in measure in Ω and lim k→+∞ ρ λ (u k ) = ρ(u). 9 [...]... 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Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence and multiplicity of positive solutions. distribution, and reproduction in any medium, provided the original work is properly cited. Existence and multiplicity of positive solutions for a class of p(x)-Kirchho type equations Ruyun Ma, Guowei Dai ∗ and