Least energy sign-changing solutions for a class of nonlocal Kirchhoff-type problems

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When b > 0, problem (1) is involving the term b |∇u| 2 dx and this term makes (1) a nonlocal problem. Such kind of problems is related to the stationary analogue of the Kirchhoff equation (1) − a + b � |∇u| 2 dx �u = g(x, u), x ∈ �, u = 0, x ∈ ∂�, proposed by Kirchhoff (1883) as an extension of the classical D' Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. For more mathematical and physical background on Kirchhoff type problems, we refer the readers to Chipot and Lovat (1997).
In the recent years, many authors have also studied the following Kirchhoff type problems defined on the whole space R N where V ∈ C(R N , R) and h ∈ C(R N × R, R). There are many interesting studies on the existence and multiplicity of solutions to problem (1) and (2), see for example, He and Zou (2012), Mao andZhang (2009), Zhang andPerera (2006) and the reference therein.
Next, we give some notations. Let X := H 1 0 (�) be the Sobolev space equipped with the inner product and the norm Throughout the paper, let X ′ denote the dual of X and �·, ·� be the duality pairing between X ′ and X. We denote by | · | r the usual L r -norm. Since is a bounded domain, it is well known that X ֒→ L r (�) continuously for r ∈ [1, 2 * ], compactly for r ∈ [1, 2 * ). Hence, for r ∈ [1, 2 * ], there exists γ r such that Recall that a function u ∈ X is called a weak solution of (1) if Seeking a weak solution of problem (1) is equivalent to finding a critical point of C 1 functional Moreover, We call u ∈ X is a sign-changing solution of (1), if u ∈ X is a solution of (1) and u ± � = 0, where u + (x) = max{u(x), 0} and u − (x) = min{u(x), 0}.
However, regarding on the existence of sign-changing solutions of problem (1), to the best of knowledge, there are very few results in the context. Recently, Mao and Zhang (2009), Zhang and Perera (2006) studied the existence of one sign-changing solution (3) |u| r ≤ γ r �u�, ∀ u ∈ X.
via invariant sets of descent flow with g satisfying asymptotically 3-linear condition. Very recently, combining constraint variational methods and quantitative deformation lemma, Shuai (2015) firstly obtained the existence of one least energy sign-changing solution of problem (1) with g(x, u) = g(u) ∈ C 1 (R, R) by seeking a minimizer of the energy functional J b over the following constraint: and proved that the minimizer is a sign-changing solution of (1), which is so called least energy sign-changing solution.
Here we must point out that the most crucial ingredients of his proofs are to show M b � = ∅ by using Implicit Function Theorem, and thus g ∈ C 1 is necessary. But, in the present paper we will show g ∈ C 1 is not necessary. By using some subtle analytical skills, we can relax g ∈ C 1 to g ∈ C 0 , and still obtain the existence of the least energy sign-changing of (1).
We are now in a position to state the first main result of this paper.
Theorem 1 Assume that conditions (g 1 )-(g 4 ) hold. Then problem (1) has one least energy sign-changing solution u b ∈ M b , which has two nodal domains.
Remark 2 Compared with Theorem 1.1 in Shuai (2015), we only need g ∈ C 0 not C 1 to ensure the existence of least energy sign-changing solutions for (1). Hence our Theorem 1 generalizes his result to more general nonlinearity.
When g ∈ C 1 , Shuai (2015) compared the energy of any sign-changing solutions with the ground state energy of (1). He obtained the energy of any sign-changing solutions is larger than that of the ground state solutions of (1), and claimed whether the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1) or not was unknown. In the present paper, we will give an affirmative answer that (1) has the property of the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1), which is called energy doubling property by Weth (2006). Precisely, we establish the second main result as follows.
Theorem 3 In addition that g ∈ C 1 in Theorem 1, then 0 < c b is the ground state energy corresponding to the ground state solution v b ∈ X of (1), and where u b is the least energy sign-changing solution of (1) obtained in Theorem 1, and Remark 4 Since c b > 0, it follows from (4) that the ground state solution v b of (1) is either a positive or a negative function in X, and (1) has energy doubling property. Hence, our Theorem 3 improves Theorem 1.2 in Shuai (2015).

Proof of main results
We assume that (g 1 )-(g 4 ) are satified from now on. In order to seek the least energy signchanging solutions of (1), the most crucial ingredient of the proof is to show M b � = ∅ .
To begin with, for any fixed u ∈ X with u ± � = 0, we consider a function J u defined on Next, we further give the following properties of J u .
Lemma 5 For any fixed u ∈ X with u ± � = 0, J u has a unique critical point (s u , t u ) with s u , t u > 0, which is the unique maximum point of J u on R + × R + .
Proof For any ǫ > 0, by (g 1 ) and (g 2 ), there exists C ǫ > 0 such that Moreover, for any M > 0, from (g 3 ), (g 4 ) and (6), there exists C M > 0 such that In order to obtain the desired results, next we divide the proof into three steps.
Next we show that the property (ii) holds for δ(t). Arguing by contradiction, if there exists {t n } ⊂ R + with t n → +∞ such that δ(t n ) ≥ t n for all n ∈ N and δ(t n ) → +∞ as n → ∞. Applying (13) again, it gives a contradiction. Hence the desired property (ii) holds.
(15) a�u + � 2 + bδ(t n ) 2 �u + � 4 + bB u t 2 as 2 0 �u + � 2 + bs 4 0 �u + � 4 + bB u s 2 0 t 2 0 = � g(s 0 u + )s 0 u + dx, Note that u ∈ M b , hence and Without loss of generality, we may assume that 0 < s 0 ≤ t 0 , then the combination of (16) and (18) implies that If s 0 < 1, by by (g 4 ), then the left hand of (20) is greater than 0, and the right hand is less than or equal to 0, which is also absurd. Hence, s 0 ≥ 1. On the other hand, in view of (17) and (19), it has If t 0 > 1, by (g 4 ), then the left hand of (21) is less than 0, and the right hand is greater than or equal to 0, which is absurd. Hence, t 0 ≤ 1. Therefore, s 0 = t 0 = 1. Consequently, the pair (1, 1) is a unique critical point of J u on (0, +∞) × (0, +∞) in the case that u ∈ M b . Case 2: u � ∈ M b . By the step 1, we have known that J u has critical point (s u , t u ) on (0, +∞) × (0, +∞). Assume that (s ′ u , t ′ u ) also be a critical point of J u on (0, +∞) × (0, +∞). Hence So, Note that v 1 ∈ M b , from the Case 1, (22) implies that s ′ u s u = t ′ u t u = 1. Hence s ′ u = s u and t ′ u = t u , which implies that the pair (s u , t u ) is a unique critical point of J u on (0, +∞) × (0, +∞) in the case that u � ∈ M b .
Step 3 (s u , t u ) is the unique maximum point of J u on R + × R + . The proof is same to the Lemma 2.3 in Shuai (2015), so we omit it here. This completes the proof.
Remark 6 Throughout of the proof, making use of some subtle analytical skills instead of Implicit Function Theorem used in Shuai (2015), we only need g(x, u) ∈ C 0 (� × R, R) not g(x, u) = g(u) ∈ C 1 (R, R) which is independent in x in Shuai (2015). Hence, we greatly relax constraints on g.
From Lemma 5, we directly deduce the following Corollary 2.3, which is crucial for comparing the energy of any sign-changing solutions with that of the ground state solutions of (1).  (2015), the rest proof can be derived by some slightly modifications of the proof of Theorem 1.1 in Shuai (2015). But we must point out that it only needs g(x, u) ∈ C 0 (� × R, R) throughout of the proof.

Corollary 7 If
In order to establish the property of the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1), we also need the following lemma.
Lemma 8 For any fixed u ∈ X\{0}, there exists a unique λ u > 0 such that λ u u ∈ N b .
Proof We consider the function By (6) and (7), we conclude that φ(λ) > 0 for λ > 0 small and φ(λ) < 0 for λ > 0 large. Then the continuity of φ(λ) implies there exists λ u > 0 such that φ(λ u ) = 0, i.e., Without loss of generality, we may assume λ u < λ ′ u , it follows from (23), (24) and (g 4 ) that which is a contradiction. Hence the uniqueness of λ u holds and the proof is completed. □ J u (s, t), Proof of Theorem 3 Note that g(x, u) ∈ C 1 (� × R, R), then N b is manifold of C 1 and the critical points of the functional J b on N b are critical points of J b on X due to Corollary 2.9 in He and Zou (2012). Similarly to the proof of Theorem 1.2 in Shuai (2015), we can prove the existence of the ground state solution is the least energy sign-changing solutions of (1) obtained in Theorem 1, by Lemma 8, there exists a unique pair (s Hence, it follows from Corollary 7 that Hence, (1) has the property of the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1) and the proof is completed.

Conclusion
On the one hand, using some subtle analytical skills and relaxing g ∈ C 1 in Shuai (2015) to g ∈ C 0 , the existence of the least energy sign-changing solutions of (1) is also obtained successfully. On the other hand, we give an affirmative answer that the energy of any sign-changing solutions is larger than twice that of the ground state solutions of (1). Hence, Our results generalize and improve Theorems 1.1 and 1.2 in Shuai (2015), respectively.