The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems \begin{document} $ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ), $ \end{document} where $a, b>0$, $\lambda 0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 21 ], X.H. Tang and B.T. Cheng (2016)[ 22 ].

1. Introduction and and main results. In this paper, we are concerned with the existence and nonexistence of ground state nodal solutions of the following Kirchhoff type problem: where Ω is a smooth bounded domain in R N , N = 1, 2, 3, a, b are positive constants, λ < aλ 1 , λ 1 is the principal eigenvalue of (− , H 1 0 (Ω)). Problem (1) is called nonlocal because of the presence of the term which implies that the equation in (1) is no longer a point-wise identity. This phenomenon provokes some mathematical difficulties, which make the study of such a class of problems particularly interesting. Problem (1) is related to the stationary analogue of the equation proposed by Kirchhoff [12] in 1883 as a generalization of the well-known D'Alembert wave equation for free vibrations of elastic strings. We have to point out that nonlocal problems also appear in other fields, such as biological systems, please refer to [1,5]. However, problem (2) received great attention only after Lions [15] proposed an abstract functional analysis framework for the problem. When b = 0 in problem (1), it reduces to the classic semilinear elliptic problem. Bartsch, Weth and Willem [4] have obtained a ground state nodal solution. After that many authors are devoted to the investigations for a variety of elliptic equations on bounded domain or whole space. Remarkably, (1) is a nonlocal problem which causes that the energy functional has totally different properties from the case b = 0, which makes the study of problem (1) particularly interesting.
Kirchhoff type problems have been paid much attention to various authors, especially on the existence of positive solutions, multiple solutions, ground state solutions, semiclassical states and the concentration behavior of positive solutions, see for example, [10,11,13,14,16,20,23,26] and the references therein. However, regarding the existence of nodal solutions for the following Kirchhoff type problem to the best of our knowledge, there are a few results in the context, such as [8,9,17,18,19,21,22,25,28]. For the case that f satisfies asymptotically 3-linear growth condition, Zhang and Perera [28], Mao and Luan [18] studied the existence of one nodal solution via invariant sets of descent flow. For the case that the nonlinearity f satisfies super-3-linear growth condition, by constraint variational methods and the quantitative deformation lemma, Figueiredo and Nascimento [9] studied the existence of ground state nodal solutions for problem (3), where f satisfies the (AR)-condition: 0 < θ s 0 f (t)dt ≤ f (s)s for some θ ∈ (4, 6), ∀|s| > 0. After that, Shuai [21] studied the existence and asymptotic behavior of ground state nodal solutions for problem (3), where f ∈ C 1 (R, R) satisfies the following conditions: s p−1 = 0 for some constant 4 < p < 2 * , where 2 * = +∞ for N = 1, 2 and 2 * = 6 for N = 3; ( |s| 3 is an increasing function of s ∈ R\{0}. Recently, Tang and Cheng [22] improved and generalized some results of Shuai [21] by replacing the monotonicity condition (f 4 ) with the weaker condition GROUND STATE NODAL SOLUTIONS 613 (f 5 ) there exists a θ 0 ∈ (0, 1) such that for all t > 0 and τ ∈ R\{0}, We must point out that Shuai [21], Tang and Cheng [22] studied the existence of ground state nodal solutions to problem (3) when f satisfies super-3-linear growth condition at infinity and superlinear growth at zero. So, a natural question is whether these conditions can be relaxed to obtain the same results. Motivated by the previously mentioned works, in the present paper, we shall consider the case f satisfies 3-linear growth condition at infinity and linear growth at zero, in other words, we will investigate the existence and nonexistence of ground state nodal solutions to problem (1) and give a convergence property of ground state nodal solutions as b 0. Besides, we firstly establish the nonexistence results of nodal solutions to problem (1).
When dealing with problem (1), we delicately analyze the behaviors of the term b Ω |∇u| 2 dx u and the term |u| 2 u, and find that both b Ω |∇u| 2 dx 2 and Ω u 4 dx are 4-order. This observation indicates that problem (1) does not always have nodal solutions or even solutions for all b > 0. Hence, we divide the issue into two circumstances and give the existence and nonexistence results of ground state nodal solutions to problem (1). On the other hand, this observation also indicates that the methods used in above papers cannot be used here directly. Indeed, we give a more refined analysis about the existence and nonexistence of ground state nodal solutions to problem (1). Next, we give some notations. Throughout this paper, let H 1 0 (Ω) be the usual Sobolev space equipped with the norm u = Ω |∇u| 2 dx 1 2 , | · | s be the usual Lebesgue space L s (Ω) norm. Let S 2 be the principal eigenvalue of the following nonlinear eigenvalue problem It is well known that S is obtained by an associated eigenfunction e 1 which is strictly positive in Ω by [28]. In particular, S is defined as Define an energy functional J b on the space H 1 0 (Ω) by Then J b is well defined on H 1 0 (Ω) and is of C 1 , and for each u, v ∈ H 1 0 (Ω), we have where ·, · denotes the usual duality. It is standard to verify that the weak solutions of problem (1) correspond to the critical points of the functional J b . Furthermore, if u ∈ H 1 0 (Ω) is a solution of problem (1) and u ± = 0, then u is a nodal solution of problem (1), where u + (x) := max{u(x), 0} and u − (x) := min{u(x), 0}.

XIAO-JING ZHONG AND CHUN-LEI TANG
Here a solution is called a ground state (or least energy) nodal one if it possesses the least energy among all nodal solutions. By a simple calculation, we can obtain that When b = 0, problem (1) does not depend on the nonlocal term Ω |∇u| 2 dx u any more, i.e., it becomes which corresponds to the energy functional J 0 : Similarly, J 0 is well defined on H 1 0 (Ω) and is of C 1 , and From (5),(6), (7), it is easy to see that there are some essential differences in studying the nodal solutions for problem (1) between b > 0 and b = 0 because the so-called nonlocal term b Ω |∇u| 2 dx u. Therefore, the methods of seeking nodal solutions for problems as (8) seem to be not applicable to problem (1). Inspired by the above mentioned works, we will consider the following minimization problems: , whose minimizers are corresponding to the nodal solutions for problems (1) and (8), respectively.
Another aim of the paper is to show the energy of any nodal solutions of problem (1) is strictly larger than twice that of the ground state solutions of problem (1), and establish the convergence of the ground state nodal solution as b 0. As usual, we seek the ground state solutions of problems (1) and (8) as minimizers of corresponding energy functionals J b and J 0 on the following Nehari manifolds: Our main results can be stated as follows. (1) has at least one ground state nodal solution which has precisely two nodal domains, and (1) does not admit any nodal solution, and Λ ≤ 1 2S 2 . Theorem 1.2. For each λ < aλ 1 , for any sequence {b n } with b n 0 as n → ∞, there exists a subsequence, still denoted by {b n }, such that u bn convergent to u 0 strongly in H 1 0 (Ω), where u 0 is a ground state nodal solution of problem (8) which has precisely two nodal domains. Remark 1. Our results make good explanation for the existence and nonexistence of ground state nodal solutions to problem (1) if N = 1, 2, 3. However, if N = 4, problem (1) involves the critical nonlinearity |u| 2 u because 2 * = 4. As far as we know, Daisuke Naimen [20] proved that if 0 < λ < aλ 1 , problem (1) Comparing with [21] and [22], we investigate the existence of ground state nodal solutions to problem (1) and give a convergence property of ground state nodal solutions as b 0 when the nonlinearity satisfies 3-linear growth condition at infinity and linear growth at zero. However, Shuai [21], Tang and Cheng [22] considered the case the nonlinearity satisfies super-3-linear growth condition at infinity and superlinear growth at zero. Since both Ω |∇u| 2 dx 2 and Ω u 4 dx are 4-order, we introduce some new ideas to prove that M b = ∅. Moreover, we firstly give the nonexistence result of nodal solutions to problem (1). Consequently, our results can be regarded as the extension and supplementary work of [21] and [22].
We organize this paper as follows. In Section 2 we present some notations and prove some useful preliminary lemmas which pave the way for getting one ground state nodal solution. Then Section 3 is devoted to proving Theorem 1.1 and Theorem 1.2, and obtaining the existence and nonexistence results of nodal solutions.
2. Some preliminary lemmas. In this section, we give some preliminary lemmas which are crucial for proving our results. Firstly, we will check that there is a unique pair (s u , t u ) of positive numbers such that s u u Proof. Let λ < aλ 1 , u ∈ H 1 0 (Ω) with u ± = 0 and (9), then su

XIAO-JING ZHONG AND CHUN-LEI TANG
Hence, we only need to show that there is only one positive solution (S, T ) to the following system It is easy to see from (9) that Consequently, Together with λ < aλ 1 , we have a u ± 2 > λ Ω |u ± | 2 dx and Let S = D S D and T = D T D , then (S, T ) ∈ (0, +∞) × (0, +∞) is the unique solution to system (10). Choosing s u = √ S and t u = √ T , we can obtain that (s u , t u ) is the unique pair of positive numbers such that s u u Furthermore, since it is not difficult to verify that From the fact that (s 2 u , t 2 u ) is the solution of system (10), we have We consider the Hessian matrix of .
Proof. Let u ∈ M b , we have from the definition of M b that u ± = 0 and Since λ < aλ 1 and λ 1 is the principal eigenvalue of (− , H 1 0 (Ω)), we can obtain that Then we have completed the proof. Lemma 2.3. Assume that λ < aλ 1 , u ∈ H 1 0 (Ω) with u ± = 0 and J b (u), u ± ≤ 0, then there is a unique pair (s u , t u ) ∈ (0, 1] × (0, 1] such that Since λ < aλ 1 , it is clear that a u ± 2 > λ Ω |u ± | 2 dx and Then by Lemma 2.1, there is a unique pair (s u , t u ) of positive numbers such that It means that (s 2 u , t 2 u ) is the solution of system (10). Similar to the argument of Lemma 2.1, we have from (17) that Therefore, Lemma 2.4. If λ < aλ 1 , for any u ∈ H 1 0 (Ω) with b u 4 < Ω u 4 dx, there exists a uniques u > 0 such thats u u ∈ N b . Moreover, J b (s u u) > J b (su) for all s ≥ 0 and s =s u .
Proof. If λ < aλ 1 , for any u ∈ H 1 0 (Ω) with b u 4 < Ω |u| 4 dx, one can get that su ∈ N b if and only if as 2 u 2 + bs 4 u 4 = λs 2 Ω u 2 dx + s 4 Ω u 4 dx, it is easy to see that there exists a uniques u = a u 2 −λ Ω u 2 dx we have J b (s u u) > J b (su) for all s ≥ 0 and s =s u .
then the following results hold.
we only need to show that for all 0 < b < 1 S 2 , there exists at lease one u ∈ H 1 0 (Ω) with b u 4 < Ω u 4 dx by Lemma 2.4. In fact, we can pick up the extremal function e 1 for S by (4), then for each 0 It follows that e 1 is the one that meets the requirements, which means that N b = ∅. For each u ∈ N b , it follows from λ < aλ 1 , 0 < b < 1 S 2 and Sobolev inequality (4) that Therefore, and J b is coercive and bounded below on N b for all λ < aλ 1 and 0 < b < 1 S 2 . Let {v n } ⊂ N b is a minimizing sequence for J b . Obviously, J b (v n ) = J b (|v n |) and |v n | ∈ N b and therefore we can assume from the beginning that v n (x) ≥ 0 a.e. in Ω and for all n. It follows from the fact J b is coercive on N b that the sequence {v n } is bounded in H 1 0 (Ω), so that, up to subsequences, v n v b in H 1 0 (Ω) and v b (x) ≥ 0. We now prove that v n → v b strongly in H 1 0 (Ω). Supposing the contrary, then v b < lim inf n→∞ v n , we get Similar to the argument in Brown and Zhang [7], we can conclude v b is a critical point of J b . And by the strong maximum principle, u is strictly positive.
(ii) For each u ∈ H 1 0 (Ω) with u ± = 0, (9) holds for small enough b > 0, which means from Lemma 2.1 that M b = ∅. Thus We first verify that M b = ∅ for all 0 < b < Λ. Obviously, it follows from the definition of Λ that M Λ−ζ = ∅ for sufficiently small ζ > 0. Therefore, by the fact that ζ is arbitrary, it suffices to show that Thus, for all 0 < b < Λ − ζ, we have that It follows from Lemma 2.1 that there is a pair (s b , t b ) of positive numbers such that Then similarly, we can obtain that Thirdly, assume that {u n } ⊂ M b is a minimizing sequence for J b , namely such that J b (u n ) → m b . We have already observed that J b is coercive on N b , this implies that the sequence {u n } is bounded in H 1 0 (Ω), going if necessary to a subsequence, still denoted by {u n }, we can assume that there exists a u b ∈ H 1 0 (Ω) such that, for n sufficiently large, Passing to the limit, we obtain from (18) and λ < aλ 1 that Then by Lemma 2.3, there is a unique pair (s u , t u ) ∈ (0, 1] × (0, 1] such that Lastly, if J b (u b ) = 0, there exist δ > 0 and α > 0 such that It follows from Lemma 2.1 that By Lemma 2.1 and (b), we obtain that max (s,t)∈D We prove that η(1, ψ(D)) ∩ M b = ∅, contradicting to the definition of m b . Let us define γ(s, t) = η(1, ψ(s, t)) and γ(s, t)), γ + (s, t) , J b (γ(s, t)), γ − (s, t) .
3. Proof of the main results. In this section, we will prove the main results. To begin with, we will show that the ground state nodal solution u b of problem (1) has precisely two nodal domains, and its energy is strictly larger than twice that of the ground state energy. < Ω |u ± b | 4 dx, it follows from Lemma 2.4 that there exist s 1 , t 1 > 0 such that . Now, we show that u b has exactly two nodal domains. We assume by contradiction that u b = u 1 + u 2 + u 3 with Moreover, using the fact that J b (u b ) = 0, we get Consequently, by Lemma 2.3, there exist (s,t) ∈ (0, 1] × (0, 1] such that Noting that λ < aλ 1 , J b (u b ), u b = 0 and J b (su 1 +tu 2 ),su 1 +tu 2 = 0, we have which leads to a contradiction, and thus the minimizer u b has precisely two nodal domains.
(ii) Let λ < aλ 1 and b ≥ Λ, we claim that problem (1) Since D is continuous at Λ for the fixed u, we can deduce that there exists κ > 0 such that for all b ∈ (Λ − κ, Λ + κ), Then by the proof of Lemma 2.1, one finds that system (10) possesses one positive solution, denoted by (S b , T b ), and thus which contradicts the definition of Λ. Therefore, problem (1) does not admit any nodal solution for all λ < aλ 1 and b ≥ Λ.
Furthermore, we will show that M b = ∅ for all b ≥ 1 2S 2 , which means that Λ ≤ 1 2S 2 . If b ≥ 1 2S 2 , there exists u ∈ M b , we have from Lemma 2.2 that (9) holds. Then it follows from Sobolev inequality (4) and b ≥ 1

XIAO-JING ZHONG AND CHUN-LEI TANG
Combining with (20) and the first inequality of (9), one can get that Then we can obtain from u + = 0 that On the other hand, by Sobolev inequality (4) and b ≥ 1 2S 2 , we can easily get that Similarly, we deduce from (22) and the second inequality of (9) that then u + 2 < u − 2 , which contradicts (21). Hence, M b = ∅ for all b ≥ 1 2S 2 ,and we have proved that Λ ≤ 1 2S 2 . Now, we are in a situation to prove Theorem 1.2. In the following, we regard b > 0 as a parameter in problem (1). We shall analyze the convergence property of u b as b 0.
Proof of Theorem 1.2. For any b 0, let u b ∈ M b be the ground state nodal solution to problem (1), which changes sign only once.
Choose a nonzero function w 0 ∈ C ∞ 0 (Ω) such that w ± 0 = 0, there exists a 0 < β < Λ such that Thus, for any b ∈ [0, β], it follows from λ < aλ 1 and (23) that It is not difficult to see that there exists θ > 0 such that for all s, t > 0, where θ = max s,t≥0 h(s, t) and For any sequence {b n } with b n 0 as n → ∞, one can obtain from Theorem 1.1 that for large n, there exists u bn ∈ M bn is a nodal critical point of J bn , then This shows that {u bn } is bounded in H 1 0 (Ω), then there exists a subsequence of {b n }, still denoted by {b n }, such that u bn u 0 weakly in H 1 0 (Ω). By the compactness of the embedding H 1 0 (Ω) → L s (Ω) for 1 ≤ s < 6, using a standard argument, we can prove that u ± bn → u ± 0 strongly in H 1 0 (Ω), and u ± 0 = 0. Furthermore, we deduce that for all u ∈ H 1 0 (Ω), In the proof of Theorem 1.1, b = 0 is allowed. Then there exists a v 0 ∈ M 0 such that and v 0 is a nodal solution to problem (1) which changes sign only once. Similarly, we can pick up ∈ (0, Λ) which is independent on b n such that In the same way, we can obtain that and by J 0 (v 0 ), v ± 0 = 0, we have Combining with (25) and (26), one has s n → 1, t n → 1 as n → ∞. Now, we only need to show J 0 (u 0 ) = J 0 (v 0 ), then by (24), u 0 is a ground state nodal solution of problem (8) which changes sign only once. In fact, . This completes the proof of Theorem 1.2.