Abstract
In this article, we study the nonlocal p(x)Laplacian problem of the following form
where Ω is a smooth bounded domain and ν is the outward normal vector on the boundary ∂Ω, and . By using the variational method and the theory of the variable exponent Sobolev space, under appropriate assumptions on f, g, a and b, we obtain some results on existence and multiplicity of solutions of the problem.
Mathematics Subject Classification (2000): 35B38; 35D05; 35J20.
Keywords:
critical points; p(x)Laplacian; nonlocal problem; variable exponent Sobolev spaces; nonlinear Neumann boundary conditions1 Introduction
In this article, we consider the following problem
where Ω is a smooth bounded domain in R^{N}, with 1 < p^{ }:= inf_{Ω }p(x) ≤ p(x) ≤ p^{+ }:= sup_{Ω }p(x) < N, a(t) is a continuous realvalued function, f : Ω × R → R, g : ∂Ω × R → R satisfy the Caratheodory condition, and . 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
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 functional analysis framework for the problem was proposed; see e.g. [36] 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 nonlocal 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 pLaplacian; 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
The study of the stationary version of Kirchhoff type problems has received considerable attention in recent years; see e.g. [5,1116].
The operator Δ_{p(x)}u = div(∇u^{p(x)2}∇u) is called p(x)Laplacian, which becomes pLaplacian when p(x) ≡ p (a constant). The p(x)Laplacian possesses more complicated nonlinearities than pLaplacian. The study of various mathematical problems with variable exponent are interesting in applications and raise many difficult mathematical problems. We refer the readers to [1723] for the study of p(x)Laplacian equations and the corresponding variational problems.
Corrêa and Figueiredo [13] presented several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the pKirchhoff type equation. Fan and Zhang [20] studied p(x)Laplacian equation with the nonlinearity f satisfying AmbrosettiRabinowitz 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 nonlinear boundary conditions see [2630], for elliptic systems with nonlinear boundary conditions see [31,32], for the pLaplacian with nonlinear boundary conditions of different type see [3337], and for the p(x)Laplacian with nonlinear boundary conditions see [3840]. Motivated by above, we focus the case of nonlocal p(x)Laplacian problems with nonlinear Neumann boundary conditions. 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
we can introduce the norm on L^{p(x) }(Ω) by
and (L^{p(x) }(Ω),  · _{p(x)}) becomes a Banach space, we call it the variable exponent Lebesgue space.
The space W^{1, p(x)}(Ω) is defined by
and it can be equipped with the norm
where ∇u_{p(x) }= ∇u_{p(x)}; and we denote by the closure of in W^{1, p(x)}(Ω), , , when p(x) < N, and p* = p_{* }= ∞, when p(x) > N.
Proposition 2.1 [22,41]. (1) If , 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
(2) If , p_{1 }(x) ≤ p_{2 }(x), for any x ∈ Ω, then , and the imbedding is continuous.
Proposition 2.2 [22]. If f : Ω × R → R is a Caratheodory function and satisfies
where , , d(x) ≥ 0 and e ≥ 0 is a constant, then the superposition operator from to defined by (N_{f }(u)) (x) = f (x, u (x)) is a continuous and bounded operator.
Proposition 2.3 [22]. If we denote
then for u, u_{n }∈ L^{p(x) }(Ω)
(1) u (x)_{p(x) }< 1(= 1; > 1) ⇔ρ (u) < 1(= 1; > 1);
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 , then and W^{1,p(x)}(Ω) are separable reflexive Banach spaces;
(2) if and q (x) < p* (x) for any , then the imbedding from W^{1, p(x)}(Ω) to L^{q(x) }(Ω) is compact and continuous;
(3) if and q (x) < p_{* }(x) for any , 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
So, ∇u_{p(x) }is a norm equivalent to the norm  u  in the space .
3 Coercive functionals
In this and the next sections we consider the nonlocal p(x)LaplacianNeumann problem (P), where a and b are two real functions satisfying the following conditions
(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_{l}) f : Ω × R → R satisfies the Caratheodory condition and there exist two constants C_{1 }≥ 0, C_{2 }≥ 0 such that
where and q_{1 }(x) < p* (x), .
(g_{1}) g : ∂Ω × R → R satisfies the Caratheodory condition and there exist two constants such that
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
Lemma 3.1. Let (f_{1}), (g_{1}) (a_{1}) and (b_{1}) hold. Then the following statements hold true:
(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
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 semicontinuous, Φ, Ψ: X → R are sequentially weakly continuous, and thus E is sequentially weakly lower semicontinuous.
(4) The mappings Φ' and Ψ' are sequentially weaklystrongly continuous, namely, u_{n }⇀ u in X implies Φ' (u_{n}) → Φ' (u) in X*. For any open set D ⊂ X\{0} with , The mappings J' and are bounded, and are of type (S_{+}), namely,
Definition 3.1. Let c ∈ R, a C^{1}functional E : X → R satisfies (P.S)_{c }condition 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 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 for t ≥ M.
(b_{2}) There are positive constants β_{1 }and C such that 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
and hence E is coercive. Since E is sequentially weakly lower semicontinuous 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:
(a_{3}) There is a positive constant α_{2 }such that .
(b_{3}) There is a positive constant β_{2 }such that .
(f_{2}) There exist an open subset Ω_{0 }of Ω and r_{1 }> 0 such that uniformly for x ∈ Ω_{0}.
(g_{2}) There exists r_{2 }> 0 such that uniformly for x ∈ ∂Ω.
(H_{2}) β_{2}r_{1 }< α_{2 }p_{}, r_{2 }< α_{2 }p_{}.
Then (P) has at least one nontrivial solution which is a global minimizer of the energy functional E.
Proof. From Theorem 3.1 we know that E has a global minimizer u_{0}. It is clear that F (x, 0) and consequently E (0) = 0. Take . Then, by (f_{2}), (g_{2}) (a_{3}), (b_{3}) and (H_{2}), for sufficiently small λ > 0 we have that
Hence E (u_{0}) < 0 and u_{0 }≠ 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:
(f_{3}) f (x,  t) =  f (x, t) for x ∈ Ω and t ∈ R.
(g_{3}) g (x,  t) =  g (x, t) for x ∈ ∂Ω and t ∈ R.
Then (P) has a sequence of solutions {u_{n}} such that E(u_{n}) < 0.
Theorem 3.4. Let (f_{1}), (g_{1}), (a_{1}), (b_{1}), (a_{2}), (b_{2}), (a_{3}), (b_{3}), (H_{1}), (H_{2}) 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.
(f_{2})_{+}There exist an open subset Ω_{0 }of Ω and r_{1 }> 0 such that uniformly for x ∈ Ω_{0}.
(g_{2})_{+ }There exists r_{2 }> 0 such that uniformly for x ∈ ∂Ω.
Then (P) has at least one nontrivial nonnegative solution with negative energy.
Proof. Define
Then, using truncation functions above, similarly in the proof of Theorem 3.4 in [25], we can prove that has a nontrivial global minimizer u_{0 }and u_{0 }is a nontrivial nonnegative solution of (P).
4 The Mountain Pass theorem
In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).
Lemma 4.1. Let (f_{1}), (g_{1}), (a_{1}), (b_{1}) and the following conditions hold true:
with α_{1}p_{ }> 1.
(a_{4}) ∃ λ > 0, M > 0 such that
(b_{4}) ∃θ > 0, M > 0 such that:
(f_{4}) ∃μ > 0, M > 0 such that:
0 ≤ μF(x, t) ≤ f(x, t)t, for t ≥ M and x ∈ Ω.
(g_{4}) ∃κ > θμ > 0, M > 0 such that:
0 ≤ κG(x, t) ≤ g(x, t)t, t ≥ M and x ∈ ∂Ω.
(H_{3}) λp_{+ }< θμ.
Then E satisfies condition (P.S)c for any c ≠ 0.
Proof. By (a_{4}), for u large enough,
From (f_{4}) and (g_{4}) we can see that there exists C_{1 }> 0 and C_{2 }> 0 such that
and thus, given any ε ∈ (0, μ), there exists M_{ε }≥ M > 0 and such that
We may assume and . Note that in this case the inequalities and are equivalent to and , because and for all u ∈ X. We claim that there exist C_{ε }> 0 and such that
Indeed, when and , the validity is obvious. When and , i.e., and , we have that
and
Now let {u_{n}} ⊂ X\{0}, E(u_{n}) → c ≠ 0 and E'(u_{n}) → 0. By (H_{3}), there exists ε > 0 small enough such that λp_{+ }< θ(μ  ε). Then, since {u_{n}} is a (P.S)_{c }sequence, for sufficiently large n, we have
Since α_{1}p_{ }> 1, we have that {u_{n}} is bounded. By Lemma 3.2, E satisfies condition (P.S)_{c }for c ≠ 0.
Theorem 4.1. Under the hypotheses of Lemma 4.1, and let the following conditions hold:
(a_{5}) There is a positive constant α_{3 }such that .
(b_{5}) There is a positive constant β_{3 }such that .
(f_{5}) There exists such that 1 < r_{1}(x) < p^{*}(x) for and uniformly for x ∈ Ω.
(g_{5}) There exists such such that 1 < r_{2}(x) < p_{*}(x) for x ∈ ∂ Ω and uniformly for x ∈ ∂ Ω.
(H_{4}) α_{3}p_{+ }< β_{3}r_{1}, α_{3}p_{+ }< r_{2}, λp_{+ }< θμ.
Then (P) has a nontrivial solution with positive energy.
Proof. Let us prove this conclusion by the Mountain Pass lemma. E satisfies condition (P.S)_{c }for c ≠ 0 has been proved in Lemma 4.1.
For u small enough, from (a_{5}) we can obtain easily that , from (b_{5}), (f_{1}) and (f_{5}) we have, and in the similar way from(g_{1}) and (g_{5}) we have . Thus by (H_{4}), we conclude that there exist positive constants ρ and δ such that E(u) ≥ for u = ρ.
Let w ∈ X\{0} be given. From (a_{4}) for sufficiently large t > 0 we have , which follows that for s large enough, where d_{1 }is a positive constant depending on w. From (f_{4}) and (f_{1}) for t large enough we have for s large enough, where d_{2 }is a positive constant depending on w. From (b_{4}) for t large enough we have for s large enough, where d_{3 }is a positive constant depending on w. From (g_{4}) and (g_{1}) for t large enough we have . Hence for any w ∈ X\{0} and s large enough, , thus by (H_{3}), We conclude that E(sw) → ∞ as s → +∞.
So by the Mountain Pass lemma this theorem is proved.
By the symmetric Mountain Pass lemma, similarly in the proof of Theorem 4.8 in [40], we have the following:
Theorem 4.2. Under the hypotheses of Theorem 4.1, if, in addition, (f_{3}) and (g_{3}) are satisfied, then (P) has a sequence of solutions {±u_{n}} such that E(±u_{n}) → +∞ as n → ∞.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
EG and PZ contributed to each part of this work equally. All the authors read and approved the final manuscript.
Acknowledgements
The authors thank the two referees for their careful reading and helpful comments of the study. Research supported by the National Natural Science Foundation of China (10971088), (10971087).
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