 Research
 Open Access
Least energy signchanging solutions for a class of nonlocal Kirchhofftype problems
 Bitao Cheng^{1, 2}Email author
 Received: 28 May 2016
 Accepted: 15 July 2016
 Published: 4 August 2016
Abstract
Keywords
 Kirchhofftype problem
 Least energy signchanging solutions
 Variational approach
Mathematics Subject Classfication
 35J20
 35J60
Introduction and main results

\((g_1)\) \(g(x,t)=o(t)\) uniformly in x as \(t\rightarrow 0\).

\((g_2)\) There exists \(p\in (4,2^{*})\) such that \(g(x,t)=o(t^{p1})\) uniformly in x as \(t\rightarrow \infty \), where \(2^{*}=6\), if \(N=3\), and \(2^{*}=+\infty \), if \(N=1,2\).

\((g_3)\) \(G(x,t)/t^4\rightarrow +\infty \) uniformly in x as \(t\rightarrow \infty \), where \(G(x,t)=\int _{0}^{t}g(x,s)ds.\)

\((g_4)\) \(g(x,t)/t^3\) is an increasing function on \((\infty ,0)\) and \((0,+\infty )\) for every \(x\in \Omega \).
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 signchanging solution \(u_b\in {\mathcal {M}}_b\), which has two nodal domains.
Remark 2
Compared with Theorem 1.1 in Shuai (2015), we only need \(g\in C^0\) not \(C^1\) to ensure the existence of least energy signchanging solutions for (1). Hence our Theorem 1 generalizes his result to more general nonlinearity.
When \(g\in C^1\), Shuai (2015) compared the energy of any signchanging solutions with the ground state energy of (1). He obtained the energy of any signchanging solutions is larger than that of the ground state solutions of (1), and claimed whether the energy of any signchanging 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 signchanging 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
Proof of main results
Next, we further give the following properties of \({\mathcal {J}}_u\).
Lemma 5
For any fixed \(u\in X\) with \(u^\pm \not =0\), \({\mathcal {J}}_u\) has a unique critical point \((s_u,t_u)\) with \(s_u,t_u>0\), which is the unique maximum point of \({\mathcal {J}}_u\) on \({{\mathbb {R}}}_+\times {{\mathbb {R}}}_+\).
Proof
Step 1 The existence of critical points for \({\mathcal {J}}_u\) on \((0,+\infty )\times (0,+\infty )\).
 (i)
\(\delta (t)\) and \(\zeta (s)\) are continuous on \({{\mathbb {R}}}_+\),
 (ii)
\(\delta (t)<t\) for t large and \(\zeta (s)<s\) for s large.
Next we show that the property (ii) holds for \(\delta (t)\). Arguing by contradiction, if there exists \(\{t_n\}\subset {{\mathbb {R}}}_+\) with \(t_n\rightarrow +\infty \) such that \(\delta (t_n)\ge t_n\) for all \(n\in {\mathbb {N}}\) and \(\delta (t_n)\rightarrow +\infty \) as \(n\rightarrow \infty \). Applying (13) again, it gives a contradiction. Hence the desired property (ii) holds.
Step 2 The uniqueness of critical point for \({\mathcal {J}}_u\) on \((0,+\infty )\times (0,+\infty )\).
From the Step 1, \({\mathcal {J}}_u\) has critical points on \((0,+\infty )\times (0,+\infty )\). We consider only two cases.
Step 3 \((s_u,t_u)\) is the unique maximum point of \({\mathcal {J}}_u\) on \({{\mathbb {R}}}_+\times {{\mathbb {R}}}_+\). The proof is same to the Lemma 2.3 in Shuai (2015), so we omit it here. This completes the proof. \(\square \)
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 \(g(x,u)\in C^0(\Omega \times {{\mathbb {R}}},{{\mathbb {R}}})\) not \(g(x,u)=g(u)\in C^1({{\mathbb {R}}},{{\mathbb {R}}})\) 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 signchanging solutions with that of the ground state solutions of (1).
Corollary 7
Proof of Theorem 1
Using Lemma 5 to replace the Lemmas 2.1 and 2.3 in Shuai (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 \(g(x,u)\in C^0(\Omega \times {{\mathbb {R}}},{{\mathbb {R}}})\) throughout of the proof.
In order to establish the property of the energy of any signchanging solutions is larger than twice that of the ground state solutions of (1), we also need the following lemma. \(\square \)
Lemma 8
For any fixed \(u\in X\backslash \{0\}\), there exists a unique \(\lambda _u>0\) such that \(\lambda _uu\in {\mathcal {N}}_b\).
Proof
Proof of Theorem 3
Note that \(g(x,u)\in C^1(\Omega \times {{\mathbb {R}}},{{\mathbb {R}}})\), then \({\mathcal {N}}_b\) is manifold of \(C^1\) and the critical points of the functional \(J_b\) on \({\mathcal {N}}_b\) 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 \(v_b\in {\mathcal {N}}_b\) for (1) with \(J_b(v_b)=c_b\). \(\square \)
Conclusion
On the one hand, using some subtle analytical skills and relaxing \(g\in C^1\) in Shuai (2015) to \(g\in C^0\) , the existence of the least energy signchanging solutions of (1) is also obtained successfully. On the other hand, we give an affirmative answer that the energy of any signchanging 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.
Declarations
Acknowlegements
The author thanks the anonymous referees for their valuable suggestions and comments. This Work is partly supported by NNSF (11571370, 11361048), YNEF (2014Z153) and YNSF (2013FD046).
Competing interests
He has no competing interests.
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Authors’ Affiliations
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