Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: a+b[u]s,pp(-Δ)psu=λ|u|q-2uln|u|2+|u|ps∗-2uinΩ,u=0inRN\Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$\end{document}where N>sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N >sp$$\end{document} with s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0, 1)$$\end{document}, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, and [u]s,pp=∬R2N|u(x)-u(y)|p|x-y|N+psdxdy,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$\end{document}ps∗=Np/(N-ps)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_s^*=Np/(N-ps)$$\end{document} is the fractional critical Sobolev exponent, Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N$$\end{document}(N≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N\ge 3)$$\end{document} is a bounded domain with Lipschitz boundary and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution ub\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_b$$\end{document}. Moreover, for any λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > 0$$\end{document}, we show that the energy of ub\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_b$$\end{document} is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as b→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \rightarrow 0$$\end{document}.


Introduction
In this paper, we are interested in the existence, energy estimates and the convergence property of the least energy sign-changing solution for the following fractional Kirchhoff problems with logarithmic and critical nonlinearity: [u] p s, p (− ) s p u = λ|u| q−2 u ln |u| 2 + |u| p * s −2 u in , u = 0 i n R N \ , (1.1) where N > sp with s ∈ (0, 1), p > 1, and |u(x) − u(y)| p |x − y| N + ps dxdy, p * s = N p/(N − ps) is the fractional critical Sobolev exponent, ⊂ R N (N ≥ 3) is a bounded domain with Lipshcitz boundary and λ is a positive parameter. We denote by (− ) s p the fractional p-Laplace operator which, up to a normalization constant, is defined as denotes the open ball of R N centered at x ∈ R N and radius ε > 0.
One of the classical topics in the qualitative analysis of PDEs is the study of existence and multiplicity properties of solutions for both the Kirchhoff problems and the fractional Kirchhoff problems under various hypotheses on the nonlinearity. In the recent past there is a vast literature concerning the existence and multiplicity of solutions for the following Dirichlet problem of Kirchhoff type (1.2) Problem (1.2) is a generalization of a model introduced by Kirchhoff [24]. More precisely, Kirchhoff proposed a model given by the equation where ρ, ρ 0 , h, E, L are constants. This nonlocal model extends the classical D'Alembert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations. Since Lions [29] introduced an abstract framework to Kirchhoff-type equations, the solvability of these nonlocal problems has been well studied in the general dimension by various authors. We refer to D'Ancona and Shibata [13] and D'Ancona and Spagnolo [14] for the global solvability of various classes of Kirchhoff-type problems. We also refer to Carrier [9,10] who used a more rigorous method to model transverse vibration via the coupled governing equation of planar vibration in order to recover the nonlinear integro partial-differential equation, in which a more general Kirchhoff function was considered. In addition, the nonlocal Kirchhoff problems of parabolic type can model several biological systems, such as population density, see for example Ghergu and Rȃdulescu [21]. For more details on mathematical theories and its applications of Kirchhoff-type problems, we refer the readers to [4,14,23,26,27,41]. Problem (1.2) is a nonlocal problem because the term b |∇u| 2 dx u appears in the left-hand side of the equation, which results that (1.2) is not a pointwise identity. Moreover, the energy functional associated to (1.2) has different properties with respect to the local case corresponding to b = 0, hence several mathematical difficulties are brought naturally out in the study of the nonlocal problems (b = 0) by means of variational methods. Recently, Fiscella and Valdinoci [19] proposed a steady-state Kirchhoff model involving the fractional Laplacian by taking into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string, see [19, Appendix A] for more details. Fractional Kirchhoff-type Laplacian problems have been studied by many authors, refer [2,3,20,28,34,36,37,48,49,51].
Here we can refer the recent monograph about nonlocal fractional problems [35]. We note that the results dealing with the problem (1.2) with critical nonlinearity are relatively scarce. The main difficulty in the study of these problems is due to the lack of compactness caused by the presence of the critical Sobolev exponent.
Recently, most of the literature deals with fractional Laplacian problems with power type nonlinearities, there are a few papers that deal with the existence and multiplicity of solutions for fractional problems involving logarithmic nonlinearity. In [15], d'Avenia et al. considered the following fractional logarithmic Schorödinger equation where ω > 0. By employing the fractional logarithmic Sobolev inequality, [15] obtained the existence of infinitely many solutions. Moreover, the regularity of solutions was also discussed in [15]. In [42], Truong studied the following problem fractional p-Laplacian equations with logarithmic nonlinearity where a is a sign-changing function. Under some assumptions on V , a and λ, [42] obtained two nontrivial solutions by using Nehari manifold approach. Very recently, Xiang, Hu and Yang in [47] considered the following Kirchhoff problems with combined nonlinearity of logarithmic and power type with θ ≥ 1, and h is a sign-changing function. When λ is sufficiently small, [47] obtained two nonnegative local least energy solutions by using Nehari manifold approach.
On the other hand, the existence of sign-changing solution of nonlinear elliptic PDEs with power nonlinearities has been studied extensively for the p-Laplacian operator as well as the fractional p-Laplacian operator. We refer the reader to see [5,6,11,12,30,44] and the references therein. Consider the nonlocal problem (1.4) For p = 2, the authors in [11], have studied the problem (1.4), where the fractional Laplacian operator is defined through spectral decomposition to obtain the sign-changing solution. The method of harmonic extension was introduced by Caffarelli and Silvestre [8] to transform the nonlocal problem in to a local problem in the half cylinder × (0, ∞), by using an equivalent definition of the fraction Laplacian operator [7]. For p ∈ (1, ∞), the problem studied by Chang et al. [12], where the authors have guaranteed the existence of a sign-changing solutions by using Nehari manifold method.
Recently, many authors pay their attention to find sign-changing solutions to problem (1.2) or similar Kirchhoff-type equations, and indeed, some interesting results were obtained. For example, Zhang and Perera [50] and Mao and Zhang [32] used the method of invariant sets of descent flow to obtain the existence of a sign-changing solution of problem (1.2). In [17], Figueiredo and Nascimento considered the following Kirchhoff equation of the type: where is a bounded domain in R 3 , M is a general C 1 class function, and g is a superlinear C 1 class function with subcritical growth. By using the minimization argument and a quantitative deformation lemma, the existence of a sign-changing solution for this Kirchhoff equation was obtained. In unbounded domains, Figueiredo and Santos Júnior [18] studied a class of nonlocal Schrödinger-Kirchhof problems involving only continuous functions. Using a minimization argument and a quantitative deformation lemma, they obtained a least energy sign-changing solution to Schrödinger-Kirchhof problems. Moreover, when the problem presents symmetry, the authors showed that it has infinitely many nontrivial solutions. It is noted that combining constraint variational methods and quantitative deformation lemma, Shuai [38] proved that problem (1.2) has one least energy sign-changing solution u b and the energy of u b strictly larger than the ground state energy. Moreover, the author investigated the asymptotic behavior of u b as the parameter b 0. Later, under some more weak assumptions on g (especially, Nehari type monotonicity condition been removed), with the aid of some new analytical skills and Non-Nehari manifold method, Tang and Cheng [40] improved and generalized some results obtained in [38].
In [16], Deng, Peng, and Shuai studied the following Kirchhoff problem: The authors obtained the existence of radial sign-changing solutions with prescribed numbers of nodal domains for Kirchhoff problem (1.6) in H 1 r (R 3 ), the subspace of radial functions of H 1 (R 3 ) by using a Nehari manifold and gluing solution pieces together, when V (x) = V (|x|), f (x, u) = f (|x|, u) and satisfies some conditions. Precisely, they proved the existence of a sign-changing solution, which changes signs exactly k times for any k ∈ N. Moreover, they investigated the energy property and the asymptotic behavior of the sign-changing solution. By using a combination of the invariant set method and the Ljusternik-Schnirelman type minimax method, Sun et al. [39] obtained infinitely many sign-changing solutions for Kirchhoff problem (1.6) when f (x, u) = f (u) and f is odd in u. It is worth noticing that, in [39], the nonlinear term may not be 4-superlinear at infinity; in particular, it includes the power-type nonlinearity |u| p−2 u with p ∈ (2,4]. In [43], the authors obtained the existence of least energy sign-changing solutions of Kirchhoff-type equation with critical growth by using the constraint variational method and the quantitative deformation lemma. For more results on sign-changing solutions for Kirchhoff-type equations, we refer the reader to [16,25,31] and the references therein.

Abstract setting and main results
To the best of our knowledge, there are no results concerning the existence of signchanging solutions for fractional Kirchhoff problems with logarithmic and critical nonlinearity. Hence, a natural question is whether or not there exist nodal solutions of problem (1.1)? The goal of the present paper is to develop a thorough qualitative analysis in this direction.
We first recall some preliminary results on the fractional Sobolev space W s, p 0 ( ) with respect to the norm u = [u] s, p . We then have that W s, p 0 ( ) is continuously and compactly embedded into the Lebesgue space L r ( ) endowed the norm |u| r = |u| r dx 1 r , p < r < p * s . Denote by S r the best constant for this embedding, that is, In particular, if S is the the best constant for the embedding W s, p For the weak solution, we mean the one satisfying the given definition. It is noticed that if u ± ≡ 0

The corresponding energy functional
Our goal in this paper is then to seek the least energy sign-changing solutions of problem (1.1). As we know, there are some very interesting studies, which studied the existence and multiplicity of sign-changing solutions for the following problem: where J is the energy functional of (2.4) given by However, if b > 0, the energy functional I λ b does not possess the same decompositions as (2.7) and (2.8). In fact, a straightforward computation yields that for u ± = 0. Therefore, the method to obtain sign-changing solutions for the local problem (2.6) do not seem applicable to problem (1.1). In this paper, we follow the approach in [5] by defining the following constrained set and considering a minimization problem of I λ b on M λ b . Indeed, by using the parametric method and implicit theorem, Shuai [38] proved M λ b = ∅ in the absence of the nonlocal term. However, the nonlocal term in problem (1.1), consisting of the biharmonic operator and the nonlocal term will cause some difficulties. Roughly speaking, compared to the general Kirchhoff type problem (1.2), decompositions (2.7) and (2.8) corresponding to I λ b are much more complicated. This results in some technical difficulties during the proof of the nonempty of M λ b . Moreover, we find that the parametric method and implicit theorem are not applicable for problem (1.1) due to the complexity of the nonlocal term there. Therefore, our proof takes a different route which is inspired by [1], namely, we make use of a modified Miranda's theorem (see [33]). We are also able to prove that the minimizer of the constrained problem is also a signchanging solution via the quantitative deformation lemma and degree theory. We can now present our first main result. Theorem 2.1 There exists λ * > 0 such that for all λ ≥ λ * , problem (1.1) has a least energy sign-changing solution u b ∈ M λ b with precisely two nodal domains such that Another goal of this paper is to establish the so-called energy doubling property (cf. [45]), i.e., the energy of any sign-changing solution of problem (1.1) is strictly larger than twice the ground state energy. For the semilinear equation problem (2.6), the conclusion is trivial. Indeed, if we denote the Nehari manifold associated to problem (2.6) by then it is easy to verify that u ± ∈ N for any sign-changing solution u ∈ W s, p 0 ( ) to problem (2.6). We can deduce that (2.11) We may point out that the minimizer of (2.10) is indeed a ground state solution of problem (2.6) and c > 0 is the least energy of all weak solutions of problem (2.6). Therefore, by (2.11), it follows that the energy of any sign-changing solution of problem (2.6) is larger than twice the least energy. When b > 0, a similar result was obtained by Shuai [38] in a bounded domain . We are also interested in that whether property (2.11) is still true for problem (1.1). To answer this question, we have the following result: It is obvious that the energy of the sign-changing solution u b obtained in Theorem 2.1 depends on b. In the following, we give a convergence property of u b as b → 0, which reflects some relationship between b > 0 and b = 0 for problem (1.1).

Theorem 2.3
For any sequence {b n } with b n → 0 as n → ∞, there exists a subsequence, still denoted by {b n }, such that {u n } converges to u 0 strongly in W s, p 0 ( ) as n → ∞, where u 0 is a least energy sign-changing solution to the following problem The plan of this paper is as follows: Sect. 2 covers the proof of the achievement of least energy for the constraint problem (1.1), Sect. 3 is devoted to the proofs of our main theorems.
Throughout this paper, we use standard notations. For simplicity, we use "→" and " " to denote the strong and weak convergence in the related function space respectively. Various positive constants are denoted by C and C i . We use ":=" to denote definitions and B r (x) := {y ∈ R N | |x − y| < r }. We denote a subsequence of a sequence {u n } n as {u n } n to simplify the notation unless specified.

Some technical lemmas
and Proof Our proof will be divided into three steps.
Step 1 For any u ∈ W s, p 0 ( ) with u ± = 0, in the following, we will prove the existence of α u and β u .

Lemma 3.2 For any u
Proof Without loss of generality, let α u ≥ β u > 0.
This fact together with (3.16), we have This is a contradiction. Thus, we conclude that α u ≤ 1. Thus, we have that 0 < α u , β u ≤ 1.
Proof By the definition of c λ b , there exists a sequence {u n } ⊂ M λ b such that Obviously, {u n } is bounded in W s, p 0 ( ). Then, up to a subsequence, still denoted by {u n }, there exists u ∈ W s, p 0 ( ) such that u n u. Since the embedding W s, p By Lemma 3.1, we have that for all α, β ≥ 0. On the one hand, the Vitali convergence theorem yields that lim n→∞ |u n | q ln |u n | 2 dx → |u| q ln |u| 2 dx. (3.19) On the other hand, since u n → u in L q ( ), we have lim n→∞ |u n | q dx → |u| q dx. (3.20) Then, by (3.19), (3.20), Brézis-Lieb lemma and the weak semicontinuity of norm, we have lim inf where That is, one has that for all α ≥ 0 and all β ≥ 0. Now, we claim that u ± = 0. In fact, since the situation u − = 0 is analogous, we just prove u + = 0. By contradiction, we suppose u + = 0. Hence, let β = 0 in (3.17) and we have that for all α ≥ 0. Case 1: In view of Lemma (3.22), we obtain u + > 0, which contradicts our supposition. If A 1 > 0, by (3.22), we have that for all α ≥ 0, which is absurd by Lemma 3.3. Anyway, we have a contradiction. Case 2: B 1 > 0. One one hand, by Lemma 3.3, there exists λ * > 0 such that where S > 0 is given by (2.2).
On the other hand, since B 1 > 0, we obtain A 1 > 0. Hence, in view of (3.22), we have that which is a contradiction. That is, we deduce that u ± = 0. Second, we prove B 1 = B 2 = 0. Since the situation B 2 = 0 is analogous, we only prove B 1 = 0. By contradiction, we suppose that B 1 > 0.
Case 1: B 2 > 0. According to B 1 , B 2 > 0 and Sobolev embedding, we obtain that A 1 , A 2 > 0. Let It is easy to see that φ(α) > 0 for α > 0 small enough and φ(α) < 0 for α < 0 large enough. Hence, by continuous of φ(α), there existsα > 0 such that Similarly, there existsβ > 0 such that Since That is, any point of (α, 0) with 0 ≤ α ≤α is not the maximizer of φ. Hence, On the other hand, it is easy to see that and Then, we have that which is a contradiction. Therefore, from the above arguments, we have that B 1 = B 2 = 0. Finally, we prove that c λ b is achieved. Since u ± = 0, by Lemma 3.1, there exist α u , β u > 0 such that Furthermore, it is easy to see that By Lemma 3.2, we obtain 0 < α u , β u ≤ 1.
Since u n ∈ M λ b , according to Lemma 3.3, we get Thanks to B 1 = B 2 = 0 and the norm in W s, p 0 ( ) is lower semicontinuous, and we have that Therefore, α u = β u = 1, and c λ b is achieved by u b := u + + u − ∈ M λ b .

Proof of Theorems
In this section, we prove our main results. First, we prove Theorem 2.1. In fact, thanks to Lemma 3.4, we just prove that the minimizer u b for c λ b is indeed a sign-changing solution of problem (1.1).
Finally, we prove that u has exactly two nodal domains. To this end, we assume by contradiction that are two connected open subsets of , and Setting v := u 1 + u 2 , we see that v + = u 1 and v − = u 2 , i.e., v ± = 0. Then, there exist a unique pair (α v , β v ) of positive numbers such that Hence Moreover, using the fact that ( According to Theorem 2.1, we know that the problem (1.1) has a least energy sign-changing solution u b , which changes sign only once when λ ≥ λ * . Let λ * * = max{λ * , λ * 1 }.
Suppose that u b = u + b + u − b . As Proof of Lemma 3.1, there exist α u + b > 0 and β u − b > 0 such that Furthermore, Lemma 3.2 implies that α u + b , β u − b ∈ (0, 1). Therefore, in view of Lemma 3.1, we have that Hence, it follows that c * > 0 cannot be achieved by a sign-changing function.
Finally, we close this section with the proof of Theorem 2.3. In the following, we regard b > 0 as a parameter in problem (1.1).
Proof of Theorem 2. 3 We shall proceed through several steps on analyzing the convergence property of u b as b → 0, where u b is the least energy sign-changing solution obtained in Theorem 2.1.
Then by virtue of Lemma 3.1, we get that, for any b ∈ [0, 1], there exists a unique pair (α ω (b), β ω (b)) ∈ (0, 1] × (0, 1] such that Thus, for any λ ∈ [0, 1], we have where C * > 0 is a constant independent of λ. So, let n → ∞, it follows that which implies that {u b n } is bounded in W for all v ∈ C ∞ 0 ( ), L(u, v) is defined by (2.5). From (4.5), (4.6) and Step 1, we find that for all ν ∈ C ∞ 0 ( ), which in turn implies that u 0 is a weak solution of problem (2.12). By a similar argument as in the proof of Lemma 3.3, we conclude that u ± 0 = 0. Therefore, we complete the proof of the Step 2.
Hence, the proof of Theorem 2.3 is completed.