Abstract
Keywords:
Nonlocal problems; Neumann problem; pKirchhoff's equation1. Introduction
In this paper, we deal with the nonlocal pKirchhoff type of problem given by:
where Ω is a smooth bounded domain in R^{N}, 1 < p < N, ν is the unit exterior vector on ∂Ω, Δ_{p }is the pLaplacian operator, that is, Δ_{p}u = div(∇u^{p−2}∇u), the function M : R^{+ }→ R^{+ }is a continuous function and there is a constant m_{0 }> 0, such that
is a continuous function and satisfies the subcritical condition:
where C denotes a generic positive constant.
Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes some mathematical difficulties which makes the study of such a problem particulary interesting. This problem has a physical motivation when p = 2. In this case, the operator M(∫_{Ω}∇u^{2}dx)Δu appears in the Kirchhoff equation which arises in nonlinear vibrations, namely
PKirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult [28] and the references therein for similar problem in several cases.
This work is organized as follows, in Section 2, we present some preliminary results and in Section 3 we prove the main results.
2. Preliminaries
By a weak solution of (1.1), then we say that a function u ε W^{1,p}(Ω) such that
So we work essentially in the space W^{1,p}(Ω) endowed with the norm
and the space W^{1,p}(Ω) may be split in the following way. Let W_{c }= 〈1〉, that is, the subspace of W^{1,p}(Ω) spanned by the constant function 1, and , which is called the space of functions of W^{1,p}(Ω) with null mean in Ω. Thus
As it is well known the Poincaré's inequality does not hold in the space W^{1,p}(Ω). However, it is true in W_{0}.
Lemma 2.1 [8] (PoincaréWirtinger's inequality) There exists a constant η > 0 such that for all z ∈ W_{0}.
Let us also recall the following useful notion from nonlinear operator theory. If X is a Banach space and A : X → X* is an operator, we say that A is of type (S_{+}), if for every sequence {x_{n}}_{n≥1 }⊆ X such that x_{n }⇀ x weakly in X, and . we have that x_{n }→ x in X.
Let us consider the map A : W^{1,p}(Ω) → W^{1,p}(Ω)* corresponding to −Δ_{p }with Neumann boundary data, defined by
We have the following result:
Lemma 2.2 [9,10]The map A : W^{1,p}(Ω) → W^{1,p}(Ω)* defined by (2.1) is continuous and of type (S_{+}).
In the next section, we need the following definition and the lemmas.
Definition 2.1. Let E be a real Banach space, and D an open subset of E. Suppose that a functional J : D → R is Fréchet differentiable on D. If x_{0 }∈ D and the Fréchet derivative J' (x_{0}) = 0, then we call that x_{0 }is a critical point of the functional J and c = J(x_{0}) is a critical value of J.
Definition 2.2. For J ∈ C^{1}(E, R), we say J satisfies the PalaisSmale condition (denoted by (PS)) if any sequence {u_{n}} ⊂ E for which J(u_{n}) is bounded and J'(u_{n}) → 0 as n → ∞ possesses a convergent subsequence.
Lemma 2.3 [11]Let X be a Banach space with a direct sum decomposition X = X_{1 }⊕ X_{2}, with k = dimX_{2 }< ∞, let J be a C^{1 }function on X, satisfying (PS) condition. Assume that, for some r > 0,
Assume also that J is bounded below and inf_{X }J < 0. Then J has at least two nonzero critical points.
Lemma 2.4 [12]Let X = X_{1 }⊕ X_{2}, where X is a real Banach space and X_{2 }≠ {0}, and is finite dimensional. Suppose J ∈ C^{1}(X, R) satisfies (PS) and
(i) there is a constant α and a bounded neighborhood D of 0 in X_{2 }such that J_{∂D }≤ α and,
(ii) there is a constant β > α such that ,
then J possesses a critical value c ≥ β, moreover, c can be characterized as
Definition 2.3. For J ∈ C^{1}(E, R), we say J satisfies the Cerami condition (denoted by (C)) if any sequence {u_{n}} ⊂ E for which J(u_{n}) is bounded and (1 u_{n}) J'(u_{n}) → 0 as n → ∞ possesses a convergent subsequence.
Remark 2.1 If J satisfies the (C) condition, Lemma 2.4 still holds.
In the present paper, we give an existence theorem and a multiplicity theorem for problem (1.1). Our main results are the following two theorems.
Theorem 2.1 If following hold:
(F_{0}) , where , η appears in Lemma 2.1;
Then the problem (1.1) has least three distinct weak solutions in W^{1,p}(Ω).
Theorem 2.2 If the following hold:
(M_{1}) The function M that appears in the classical Kirchhoff equation satisfies for all t ≥ 0, where ;
Then the problem (1.1) has at least one weak solution in W^{1,p}(Ω).
Remark 2.2 We exhibit now two examples of nonlinearities that fulfill all of our hypotheses
hypotheses (F_{0}), (F_{1}), (F_{2}) and (1.2) are clearly satisfied.
hypotheses (F_{3}), (F_{4}) and (F_{5}) and (1.2) are clearly satisfied.
3. Proofs of the theorems
Let us start by considering the functional J : W^{1,p}(Ω) → R given by
Proof of Theorem 2.1 By (F_{0}), we know that f(x, 0) = 0, and hence u(x) = 0 is a solution of (1.1).
To complete the proof we prove the following lemmas.
Lemma 3.1 Any bounded (PS) sequence of J has a strongly convergent subsequence.
Proof: Let {u_{n}} be a bounded (PS) sequence of J. Passing to a subsequence if necessary, there exists u ∈ W^{1,p}(Ω) such that u_{n }⇀ u. From the subcritical growth of f and the Sobolev embedding, we see that
and since J'(u_{n})(u_{n }− u) → 0, we conclude that
In view of condition (M_{0}), we have
Using Lemma 2.2, we have u_{n }→ u as n → ∞. □
Lemma 3.2 If condition (M_{0}), (F_{1}) and (F_{2}) hold, then .
Proof: If there are a sequence {u_{n}} and a constant C such that u_{n} → ∞ as n → ∞, and J(u_{n}) ≤ C (n = 1, 2 ···), let , then there exist v_{0 }∈ W^{1,p}(Ω) and a subsequence of {v_{n}}, we still note by {v_{n}}, such that v_{n }⇀ v_{0 }in W^{1,p}(Ω) and v_{n }→ v_{0 }in L^{p}(Ω).
For any ε > 0, by (F_{1}), there is a H > 0 such that for all u ≥ H and a.e. x ∈ Ω, then there exists a constant C > 0 such that for all u ∈ R, and a.e. x ∈ Ω, Consequently
It implies ∫_{Ω}v_{0}^{p}dx ≥ 1. On the other hand, by the weak lower semicontinuity of the norm, one has
Hence , so v_{0}(x) = constant ≠ 0 a.e. x ∈ Ω. By (F_{2}), . Hence
This is a contradiction. Hence J is coercive on W^{1,p}(Ω), bounded from below, and satisfies the (PS) condition. □
By Lemma 3.1 and 3.2, we know that J is coercive on W^{1,p}(Ω), bounded from below, and satisfies the (PS) condition. From condition (F_{0}), we know, there exist r > 0, ε > 0 such that
If u ∈ W_{c}, for u ≤ ρ_{1}, then u ≤ r, we have
If u ∈ W_{0}, then from condition (F_{0}) and (1.2), we have
Noting that
we can obtain
Choose u = ρ_{2 }small enough, such that J(u) ≥ 0 for u ≤ ρ_{2 }and u ∈ W_{0}.
Now choose ρ = min{ρ_{1}, ρ_{2}}, then, we have
If inf{J(u), u ∈ W^{1,p}(Ω)} = 0, then all u ∈ W_{c }with u ≤ ρ are minimum of J, which implies that J has infinite critical points. If inf{J(u), u ∈ W^{1,p}(Ω)} < 0 then by Lemma 2.3, J has at least two nontrivial critical points. Hence problem (1.1) has at least two nontrivial solutions in W^{1,p}(Ω), Therefore, problem (1.1) has at least three distinct solutions in W^{1,p}(Ω). □
Proof of Theorem 2.2. We divide the proof into several lemmas.
Lemma 3.3 If condition (F_{3}) and (F_{5}) hold, then is anticoercive. (i.e. we have that J(u) → ∞, as u → ∞, u ∈ R.)
Proof: By virtue of hypothesis (F_{5}), for any given L > 0, we can find R_{1 }= R_{1}(L) > 0 such that
Thus, using hypothesis (F_{3}), we have
So
Since L > 0 is arbitrary, it follows that
and so
This proves that is anticoercive. □
Lemma 3.4 If hypothesis (F_{4}) holds, then .
Proof: For a given , we can find C_{ε }> 0 such that for a.e. x ∈ Ω all u ∈ R. Then
Lemma 3.5 If condition (F_{4}) (F_{5}) hold, then J satisfies the (C) condition.
Proof: Let {u_{n}}_{n ≥1 }⊆ W^{1,p}(Ω) be a sequence such that
with some M_{1 }> 0 and
We claim that the sequence {u_{n}} is bounded. We argue by contradiction. Suppose that u → +∞, as n → ∞, we set , ∀n ≥ 1. Then v_{n} = 1 for all n ≥ 1 and so, passing to a subsequence if necessary, we may assume that
from (3.2), we have ∀h ∈ W^{1,p}(Ω)
with ε_{n }↓ 0.
In (3.3), we choose h = v_{n }− v ∈ W^{1,p}(Ω), note that by virtue of hypothesis (F_{4}), we have
So we have
Since M(t) > m_{0 }for all t ≥ 0, so we have
Hence, using the (S_{+}) property, we have v_{n }→ v in W^{1,p}(Ω) with v = 1, then v ≠ 0. Now passing to the limit as n → ∞ in (3.3), we obtain
then v = ξ ∈ R. Then u_{n}(x) → +∞ as n → +∞. Using hypothesis (F_{5}), we have f(x, u_{n}(x))u_{n}(x)  pF(x, u_{n}(x)) → ∞ for a.e x ∈ Ω.
Hence by virtue of Fatou's Lemma, we have
From (3.1), we have
From (3.2), we have
With ε_{n }↓ 0. So choosing h = u_{n }∈ W^{1,p}(Ω), we obtain
Adding (3.5) and (3.6), noting that for all t ≥ 0, we obtain
comparing (3.4) and (3.7), we reach a contradiction. So {u_{n}}in bounded in W^{1,p}(Ω). Similar with the proof of Lemma 3.1, we know that J satisfied the (C) condition. □
Sum up the above fact, from Lemma 2.4 and Remark 2.1, Theorem 2.2 follows from the Lemma 3.3 to 3.5.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions.
This study was supported by NSFC (No. 10871096), the Fundamental Research Funds for the Central Universities (No. JUSRP11118).
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