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 solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions Boundary Value Problems 2012, 2012:1 doi:10.1186/1687-2770-2012-1 Erlin Guo (guoerlin@lzu.edu.cn) Peihao Zhao (zhaoph@lzu.edu.cn) ISSN 1687-2770 Article type Research Submission date 29 August 2011 Acceptance date 4 January 2012 Publication date 4 January 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/1 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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China Corresponding author: guoerlin@lzu.edu.cn Email address: zhaoph@lzu.edu.cn Abstract In this article, we study the nonlocal p(x)-Laplacian problem of the following form                a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx   −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u  = b   Ω F (x, u) dx  f(x, u) in Ω a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx  |∇u| p(x)−2 ∂u ∂ν = g(x, u) on ∂Ω, where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and F (x, u) =  u 0 f(x, t)dt. By using the variational method and the theory of the variable exponent Sobolev 1 space, under appropriate assumptions on f, g, a and b, we obtain some results on existence and multiplicity of solutions of the prob- lem. Mathematics Subject Classification (2000): 35B38; 35D05; 35J20. Keywords: critical points; p(x)-Laplacian; nonlocal problem; vari- able exponent Sobolev spaces; nonlinear Neumann boundary condi- tions. 1 Introduction In this article, we consider the following problem (P )                a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx   −div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u  = b   Ω F (x, u) dx  f(x, u) in Ω a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx  |∇u| p(x)−2 ∂u ∂ν = g(x, u) on ∂Ω, where Ω is a smooth bounded domain in R N , p ∈ C(Ω) with 1 < p − := inf Ω p(x) ≤ p(x) ≤ p + := sup Ω p(x) < N, a(t) is a continuous real-valued function, f : Ω × R → R, g : ∂Ω × R → R satisfy the Caratheodory condition, and F (x, u) =  u 0 f(x, t)dt. Since the equation contains an integral related to the unknown u over Ω, it is no longer an identity pointwise, and therefore is often called nonlocal problem. Kirchhoff [1] has investigated an equation ρ ∂ 2 u ∂t 2 −  P 0 h + E 2L  L 0     ∂u ∂x     2 dx  ∂ 2 u ∂x 2 = 0, which is called the Kirchhoff equation. Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [2], where a 2 functional analysis framework for the problem was proposed; see e.g. [3–6] for some interesting results and further references. In the following, a key work on nonlocal elliptic problems is the article by Chipot and Rodrigues [7]. They studied nonlo cal boundary value problems and unilateral problems with several applications. And now the study of nonlocal elliptic problem has already been extended to the case involving the p-Laplacian; see e.g. [8, 9]. Recently, Autuori, Pucci and Salvatori [10] have investigated the Kirchhoff type equation involving the p(x)-Laplacian of the form u tt − M   Ω 1 p(x) |∇u| p(x) dx   p(x) u + Q(t, x, u, u t ) + f(x, u) = 0. The study of the stationary version of Kirchhoff type problems has received considerable attention in recent years; see e.g. [5, 11–16]. The operator  p(x) u = div(|∇u| p(x)−2 ∇u) is called p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than p-Laplacian. The study of various mathe- matical problems with variable exponent are interesting in applications and raise many difficult mathematical problems. We refer the readers to [17–23] for the study of p(x)-Laplacian equations and the corresponding variational problems. Corrˆea and Figueiredo [13] presented several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. Fan and Zhang [20] studied p(x)-Laplacian equation with the nonlinearity f satisfying Ambrosetti–Rabinowitz condition. The p(x)-Kirchhoff type equations with Dirichlet boundary value problems have been studied by Dai and Hao [24], and much weaker conditions have been given by Fan [25]. The elliptic problems with nonlinear boundary conditions have attracted expensive interest in recent years, for example, for the Laplacian with 3 nonlinear boundary conditions see [26–30], for elliptic systems with nonlinear boundary conditions see [31, 32], for the p-Laplacian with nonlinear boundary conditions of different type see [33–37], and for the p(x)-Laplacian with non- linear boundary conditions see [38–40]. Motivated by above, we focus the case of nonlocal p(x)-Laplacian problems with nonlinear Neumann boundary condi- tions. This is a new topics even when p(x) ≡ p is a constant. This rest of the article is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we consider the case where the energy functional associated with problem (P ) is coercive. And in Section 4, we consider the case where the energy functional possesses the Mountain Pass geometry. 2 Preliminaries In order to discuss problem (P ), we need some theories on variable exponent Sobolev space W 1,p(x) (Ω). For ease of exposition we state some basic properties of space W 1,p(x) (Ω) (for details, see [22, 41, 42]). Let Ω be a bounded domain of R N , denote C + (Ω) = {p|p ∈ C(Ω), p(x) > 1, ∀x ∈ Ω}, p + = max x∈Ω p(x), p − = min x∈Ω p(x), ∀p ∈ C(Ω), L p(x) (Ω) =  u|u is a measurable real-valued function on Ω,  Ω |u| p(x) dx < ∞  , we can introduce the norm on L p(x) (Ω) by |u| p(x) = inf{λ > 0 :  Ω     u(x) λ     p(x) dx  1} and (L p(x) (Ω), | · | p(x) ) becomes a Banach space, we call it the variable exponent 4 Lebesgue space. The space W 1,p(x) (Ω) is defined by W 1,p(x) (Ω) = {u ∈ L p(x) (Ω)||∇u| ∈ L p(x) (Ω)}, and it can be equipped with the norm u = |u| p(x) + |∇u| p(x) , where |∇u| p(x) = ∇u p(x) ; and we denote by W 1,p(x) 0 (Ω) the closure of C ∞ 0 (Ω) in W 1,p(x) (Ω), p ∗ = Np(x) N−p(x) , p ∗ = (N−1)p(x) N−p(x) , when p(x) < N, and p ∗ = p ∗ = ∞, when p(x) > N. Proposition 2.1 [22, 41]. (1) If p ∈ C + (Ω), the space (L p(x) (Ω), | · | p(x) ) is a separable, uniform convex Banach space, and its dual space is L q (x) (Ω), where 1/q(x) + 1/p(x) = 1. For any u ∈ L p(x) (Ω) and v ∈ L q (x) (Ω), we have      Ω uvdx      ( 1 p − + 1 q − )|u| p(x) |v| q (x) ; (2) If p 1 , p 2 ∈ C + (Ω), p 1 (x)  p 2 (x), for any x ∈ Ω, then L p 2 (x) (Ω) → L p 1 (x) (Ω), and the imbedding is continuous. Proposition 2.2 [22]. If f: Ω×R → R is a Caratheodory function and satisfies |f(x, s)|  d(x) + e|s| p 1 (x) p 2 (x) , for any x ∈ Ω , s ∈ R, where p 1 , p 2 ∈ C + (Ω), d ∈ L p 2 (x) (Ω), d(x)  0 and e  0 is a constant, then the superposition operator from L p 1 (x) (Ω) to L p 2 (x) (Ω) defined by (N f (u))(x) = f(x, u(x)) is a continuous and bounded operator. Proposition 2.3 [22]. If we denote ρ(u) =  Ω |u| p(x) dx, ∀u ∈ L p(x) (Ω), 5 then for u, u n ∈L p(x) (Ω) (1) |u(x)| p(x) < 1(= 1; > 1) ⇔ ρ(u) < 1(= 1; > 1); (2) |u(x)| p(x) > 1 ⇒ |u| p − p(x)  ρ(u)  |u| p + p(x) ; |u(x)| p(x) < 1 ⇒ |u| p − p(x)  ρ(u)  |u| p + p(x) ; (3) |u n (x)| p(x) → 0 ⇔ ρ(u n ) → 0 as n → ∞; |u n (x)| p(x) → ∞ ⇔ ρ(u n ) → ∞ as n → ∞. Proposition 2.4 [22]. If u, u n ∈ L p(x) (Ω), n = 1, 2, , then the following statements are equivalent to each other (1) lim k→∞ |u k − u| p(x) = 0; (2) lim k→∞ ρ(u k − u) = 0; (3) u k → u in measure in Ω and lim k→∞ ρ(u k ) = ρ(u). Proposition 2.5 [22]. (1) If p ∈ C + (Ω), then W 1,p(x) 0 (Ω) and W 1,p(x) (Ω) are separable reflexive Banach spaces; (2) if q ∈ C + (Ω) and q(x) < p ∗ (x) for any x ∈ Ω, then the imbedding from W 1,p(x) (Ω) to L q (x) (Ω) is compact and continuous; 3) if q ∈ C + (Ω) and q(x) < p ∗ (x) for any x ∈ Ω, then the trace imbedding from W 1,p(x) (Ω) to L q ( x) (∂Ω) is compact and continuous; (4) (Poincare inequality) There is a constant C > 0, such that |u| p(x)  C|∇u| p(x) ∀u ∈ W 1,p(x) 0 (Ω). So, |∇u| p(x) is a norm equivalent to the norm u in the space W 1,p(x) 0 (Ω). 3 Coercive functionals In this and the next sections we consider the nonlocal p(x)-Laplacian–Neumann problem (P ), where a and b are two real functions satisfying the following con- 6 ditions (a 1 ) a : (0, +∞) → (0, +∞) is continuous and a ∈ L 1 (0, t) for any t > 0. (b 1 ) b : R → R is continuous. Notice that the function a satisfies (a 1 ) may be singular at t = 0. And f , g satisfying (f 1 ) f : Ω × R → R satisfies the Caratheodory condition and there exist two constants C 1 ≥ 0, C 2 ≥ 0 such that |f(x, t)|  C 1 + C 2 |t| q 1 (x)−1 , ∀(x, t) ∈ Ω × R, where q 1 ∈ C + (Ω) and q 1 (x) < p ∗ (x), ∀x ∈ Ω. (g 1 ) g : ∂Ω × R → R satisfies the Caratheodory condition and there exist two constants C  1 ≥ 0, C  2 ≥ 0 such that |g(x, t)|  C  1 + C  2 |t| q 2 (x)−1 , ∀(x, t) ∈ ∂Ω × R, where q 2 ∈ C + (∂Ω) and q 2 (x) < p ∗ (x), ∀x ∈ ∂Ω. For simplicity we write X=W 1,p(x) (Ω), denote by C the general positive constant (the exact value may change from line to line). Define a(t) =  t 0 a(s)ds,  b(t) =  t 0 b(s)ds, ∀ t ∈ R, I 1 (u) =  Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx, I 2 (u) =  Ω F (x, u)dx, ∀u ∈ X, J(u) = a(I 1 (u)) = a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx  , Φ(u) =  b(I 2 (u)) =  b   Ω F (x, u) dx  and Ψ(u) =  ∂Ω G(x, u)dσ, ∀u ∈ X, E(u) = J(u) − Φ(u) − Ψ(u), ∀u ∈ X, where F(x, u) =  u 0 f(x, t)dt, G(x, u) =  u 0 g(x, t)dt. 7 Lemma 3.1. Let (f 1 ), (g 1 ), (a 1 ) and (b 1 ) hold. Then the following statements hold true: (1) a ∈ C 0 ([0, ∞)) ∩ C 1 ((0, ∞)), a(0) = 0, a  (t) = a(t) > 0;  b ∈ C 1 (R),  b(0) = 0. (2) J, Φ, Ψ and E ∈ C 0 (X), J(0) = Φ(0) = Ψ(0) = E(0) = 0. Furthermore J ∈ C 1 (X\{0}), Φ, Ψ ∈ C 1 (X), E ∈ C 1 (X\{0}). And for every u ∈ X\{0}, v ∈ X, we have E  (u)v = a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx   Ω  |∇u| p(x)−2 ∇u∇v + |u| p(x)−2 uv  dx −b   Ω F (x, u) dx   Ω f(x, u)v dx −  ∂Ω g(x, u)vdσ. Thus u ∈ X\{0} is a (weak) solution of (P ) if and only if u is a critical point of E. (3) The functional J : X → R is sequentially weakly lower semi-continuous, Φ, Ψ : X → R are sequentially weakly continuous, and thus E is sequentially weakly lower semi-continuous. (4) The mappings Φ  and Ψ  are sequentially weakly-strongly continuous, namely, u n  u in X implies Φ  (u n ) → Φ  (u) in X ∗ . For any open set D ⊂ X\{0} with D ⊂ X\{0}, The mappings J  and E  : D → X ∗ are bounded, and are of type (S + ), namely, u n  u and lim n→∞ J  (u n )(u n − u) ≤ 0, implies u n → u. . Definition 3.1. Let c ∈ R, a C 1 -functional E : X → R satisfies (P.S) c con- dition if and only if every sequence {u j } in X such that lim j E(u j ) = c, and lim j E  (u j ) = 0 in X ∗ has a convergent subsequence. Lemma 3.2. Let (f 1 ), (g 1 ), (a 1 ), (b 1 ) hold. Then for any c = 0, every 8 bounded (P.S) c sequence for E, i.e., a bounded sequence {u n } ⊂ X\{0} such that E(u n ) → c and E  (u n ) → 0, has a strongly convergent subsequence. The proof of these two lemmas can be obtained easily from [25, 40], we omitted them here. Theorem 3.1. Let (f 1 ), (g 1 ), (a 1 ), (b 1 ) and the following conditions hold true: (a 2 ) There are positive constants α 1 , M , and C such that a(t) ≥ Ct α 1 for t ≥ M. (b 2 ) There are positive constants β 1 and C such that     b(t)    ≤ C + C |t| β 1 for t ∈ R. (H 1 ) β 1 q 1+ < α 1 p − , q 2+ < α 1 p − . Then the functional E is coercive and attains its infimum in X at some u 0 ∈ X. Therefore, u 0 is a solution of (P ) if E is differentiable at u 0 . Proof. For u large enough, by (f 1 ), (g 1 ), (a 2 ), (b 2 ) and (H 1 ), we have that J(u) = a(I 1 (u)) = a   Ω 1 p(x)  |∇u| p(x) + |u| p(x)  dx  ≥ a(C 1 u p − ) ≥ C 2 u α 1 p − ,      Ω F (x, u) dx     ≤ C 3 u q 1+ , Φ(u) =  b(I 2 (u)) =  b   Ω F (x, u) dx  ≤ C 4 u β 1 q 1+ +  C 4 , Ψ(u) =      ∂Ω G(x, u)dσ     ≤ C 5 u q 2+ +  C 5 , E(u) = J(u) − Φ(u) − Ψ(u) ≥ C 2 u α 1 p − − C 4 u β 1 q 1+ − C 5 u q 2+ −  C 6 , and hence E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u 0 ∈ X. In this case E is differentiable at u 0 , then u 0 is a solution of (P ). Theorem 3.2. Let (f 1 ), (g 1 ), (a 1 ), (b 1 ), (a 2 ), (b 2 ), (H 1 ) and the following conditions hold true: 9 [...]... 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(b2 ), (a3 ), (b3 ), (H1 ), (H2 ) and the following conditions hold true: (b+ ) b(t) ≥ 0 for t ≥ 0 (f+ ) f (x, t) ≥ 0 for x ∈ Ω and t ≥ 0 (g+ ) g(x, t) ≥ 0 for x ∈ ∂Ω and t ≥ 0 (f2 )+ There exist an open subset Ω0 of Ω and r1 > 0 such that lim inf + t→0 F (x,t) tr1 > 0 uniformly for x ∈ Ω0 (g2 )+ There exists r2 > 0 such that lim inf + t→0 G(x,t) tr2 > 0 uniformly for x ∈ ∂Ω Then (P ) has at least... (2009) [25] Fan, XL: On nonlocal p(x)-Laplacian Dirichlet problems Nonlinear Anal 72, 3314–3323 (2010) [26] Chipot, M, Shafrir, I, Fila, M: On the solutions to some elliptic equations with nonlinear boundary conditions Adv Diff Eq 1, 91–110 (1996) [27] Hu, B: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition Diff Integral Equ 7(2), 301–313 (1994) 18 [28] Gianni... 0 and u0 = 0 By the genus theorem, similarly in the proof of Theorem 4.3 in [18], we have the following: Theorem 3.3 Let the hypotheses of Theorem 3.2 hold, and let, in addition, f and g satisfy the following conditions: (f3 ) f (x, −t) = −f (x, t) for x ∈ Ω and t ∈ R (g3 ) g(x, −t) = −g(x, t) for x ∈ ∂Ω and t ∈ R 10 Then (P ) has a sequence of solutions {un } such that E(un ) < 0 Theorem 3.4 Let (f1... solutions for p(x)Laplacian equations in Rn Nonlinear Anal 59, 173–188 (2004) [19] Fan, XL, Shen, JS, Zhao, D: Sobolev embedding theorems for space W k,p(x) (Ω) J Math Anal Appl 262, 749–760 (2001) [20] Fan, XL, Zhang, QH: Existence of solutions for p(x)-Laplacian Dirichlet problems Nonlinear Anal 52, 1843–1852 (2003) [21] Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of p(x)-Laplacian Dirichlet problem J Math... exists r1 ∈ C 0 (Ω) such that 1 < r1 (x) < p∗ (x) for x ∈ Ω and lim inf t→0 |F (x,t)| |t|r1 (x) < +∞ uniformly for x ∈ Ω ¯ (g5 ) There exists r2 ∈ C 0 (Ω) such that 1 < r2 (x) < p∗ (x) for x ∈ ∂Ω and lim inf t→0 |G(x,t)| |t|r2 (x) < +∞ uniformly for x ∈ ∂Ω (H4 ) α3 p+ < β3 r1− , α3 p+ < r2− , λp+ < θµ Then (P ) has a nontrivial solution with positive energy Proof Let us prove this conclusion by the Mountain . acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary. distribution, and reproduction in any medium, provided the original work is properly cited. Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions Erlin. Laplacian with 3 nonlinear boundary conditions see [26–30], for elliptic systems with nonlinear boundary conditions see [31, 32], for the p-Laplacian with nonlinear boundary conditions of different