## Existence of Schwarz Symmetric Minimizers for Fractional Kirchhoff Constrained Variational Problem

Wei Junping,, Huang Xiaomeng,, Zhang Yimin,

Center for Mathematical Sciences, Wuhan University of Technology, Wuhan 430070

 基金资助: 国家自然科学基金.  12171379

 Fund supported: the NSFC.  12171379

Abstract

In this paper, the existence and Schwarz symmetry of minimizers for a fractional Kirchhoff constrained variational problem with general nonlinear term were studied in space $H^s(\mathbb{R} ^N)$. Using symmetric decreasing rearrangement inequality and scaling technique, the relation between existence of minimizers for a fractional Kirchhoff constrained variational problem with the exponent of nonlinear term and parameter $c$ in $\int_{\mathbb{R} ^N}|u|^2{\rm d}x=c^2$ was discussed.

Keywords： Constrained variational ; Minimizer ; Fractional Kirchhoff problem ; Schwarz symmetric

Wei Junping, Huang Xiaomeng, Zhang Yimin. Existence of Schwarz Symmetric Minimizers for Fractional Kirchhoff Constrained Variational Problem. Acta Mathematica Scientia[J], 2022, 42(6): 1719-1728 doi:

## 1 引言

$$$m(c)=\inf \left\{E(u):u\in S_c\right\},$$$

$a, b>0$, $c$是给定的常数, $0<s<1$,

$H^s({{\Bbb R}} ^N)$表示通常的Besov空间, 定义为

$$$\left(a+b\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2} u|^2{\rm d}x \right)(-\Delta)^su-g(x, u)=\mu u.$$$

$(G_{0}) $$G:[0, \infty)\times {{\Bbb R}} \rightarrow {{\Bbb R}} 是Carathéodory函数且满足 \bullet$$ \forall\ t\in {{\Bbb R}}$, $G(., t):(0, \infty)\rightarrow {{\Bbb R}} $$(0, \infty) 上是可测的; \bullet 对每个 r\in [0, \infty) \setminus \Gamma , G(r, .): {{\Bbb R}} \rightarrow {{\Bbb R}} 连续, 其中 \Gamma 是一维零测集. (G_{1}) 对几乎处处的 r\geq 0 和每个 t\in {{\Bbb R}} , 均有 G(r, t)\leq G(r, |t|) 成立. (G_{2}) 对几乎处处的 r\geq 0 和每个 t\geq 0 , 0\leq G(r, t)\leq K(t^2+t^{p+2}), 其中 K>0 , 0<p<\frac{4s}{N-2s} . (G_{3})$$ \forall\ \varepsilon >0$, $\exists\ R_{0}>0 $$t_{0}>0 , 使得对几乎处处的 r\geq R_{0}$$ 0\leq t\leq t_{0}$, $G(r, t)\leq \varepsilon t^2$成立.

$(G_{4})$对每个$0 \leq r\leq R $$0\leq a\leq A , G(r, a)+G(R, A)\geq G(r, A)+G(R, a). 在给出本文结论之前, 我们介绍如下分数阶Gagliardo-Nirenberg不等式[7, 25] \begin{eqnarray} \int_{{{\Bbb R}} ^N} |u|^{p+2}{\rm d}x\leq \frac{p+2}{|\phi_p|_{2}^p}\alpha_p\beta_p \left(\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2} u|^2{\rm d}x\right)^{\frac{Np}{4s}}\left(\int_{{{\Bbb R}} ^N} |u|^2{\rm d}x\right)^{\frac{2ps-Np+4s}{4s}}, \end{eqnarray} 其中 0<p<\frac{4s}{N-2s} , \alpha_p=\frac{2s}{2ps-Np+4s} , \beta_p=\left(\frac{2ps-Np+4s}{Np}\right)^{\frac{Np}{4s}} . 上述等式成立当且仅当 u=\phi_p(x)$$ \phi_p(x)$是下述非线性分数阶方程唯一的非负径向解

$$$(-\Delta)^s u+u-|u|^{p}u=0, \ \ x\in{{{\Bbb R}} ^N}.$$$

$$$\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2} \phi_p|^2{\rm d}x=\frac{Np}{2s(p+2)-Np}\int_{{{\Bbb R}} ^N} |\phi_p|^2{\rm d}x=\frac{Np}{2s(p+2)}\int_{{{\Bbb R}} ^N} |\phi_p|^{p+2}{\rm d}x.$$$

$$$\tilde{m}_{c}=\inf\bigg \{E(u):u\in H^{s}_{+}({{\Bbb R}} ^N) , \ \int_{{{\Bbb R}} ^N}|u|^2{\rm d}x\leq c^2\bigg\}.$$$

(3) 当$\frac{4s}{N}<p<\frac{8s}{N} $$N < 4s 时, 对任意 在空间 H^s({{\Bbb R}} ^N) 上, 极小化问题(1.1) 存在Schwarz对称极小解, 且 c< c_2 , \liminf\limits_{s\rightarrow 0^+}\frac{G(r, s)}{s^{2}}=K , 则在空间 H^s({{\Bbb R}} ^N) 上, 极小化问题(1.1) 不存在非零极小解. (4) 当 p= \frac{8s}{N} , \liminf\limits_{s\rightarrow 0}\frac{G(r, s)}{s^{p+4}}=A>0$$ p> \frac{8s}{N}$, $\liminf\limits_{s\rightarrow 0}\frac{G(r, s)}{s^{p+2}}=B>0$时, 对任意$c>0$, 在空间$H^s({{\Bbb R}} ^N)$上, 极小化问题(1.1) 不存在非零Schwarz对称极小解.

## 2 预备定理

(i) 若$0<p<\frac{4s}{N}$, 则$\forall\ c>0$, 函数$f_p(t)$存在唯一的极小值点, 记为$t_1>0$, 使得$f_p(t)\geq f_p(t_1)$.

(ii) 若$p=\frac{4s}{N}$, 当$c>c_1=(\frac{aN}{K(4s+2N)})^{\frac{N}{4s}}|\phi_p|_{2}$时, 函数$f_p(t)$存在唯一的极小点$t_2=\frac{K(4s+2N)c^{\frac{4s}{N}}}{bN|\phi_p|_{2}^{\frac{4s}{N}}}-\frac{a}{b}>0$, 使得$f_p(t)\geq f_p(t_2)=-\frac{k^2(2s+N)^2c^{\frac{8s}{N}}}{bN^2|\phi_p|_{2}^{\frac{8s}{N}}}+\frac{aK(2s+N)c^{\frac{4s}{N}}}{bN|\phi_p|_{2}^{\frac{4s}{N}}}-\frac{a^2}{4b}-Kc^2; $$c\leq c_1=(\frac{aN}{K(4s+2N)})^{\frac{N}{4s}}|\phi_p|_{2} 时, 函数 f_p(t) 无极小值点. (iii) 若 \frac{4s}{N}<p<\frac{8s}{N} , 当 c\geq c_2=\left(\frac{\left(\frac{2as}{8s-Np}\right)^{\frac{8s-Np}{4s}} \left(\frac{bs}{Np-4s}\right)^{\frac{Np-4s}{4s}}}{K(p+2)\alpha_p\beta_p}{|\phi_p|_{2}^p}\right)^{\frac{1}{p+2-\frac{NP}{2s}}} 时, 函数 f_p(t) 存在唯一极小值点 t_3=\frac{2(Np-4s)a}{(8s-Np)b}>0 , 使得 而当 c< c_2 时, 函数 f_p(t) 无极小值点. (iv) 若 p=\frac{8s}{N} , 当 c> c_3=(\frac{bN^2|\phi_p|_{2}^{\frac{8s}{N}}}{K(4s+N)(4s-N)})^{\frac{N}{8s-2N}} 时, 函数 f_p(t) 在极大值点 t_4=\frac{aN^2|\phi_p|_{2}^{\frac{8s}{N}}}{K(4s+N)(4s-N)c^{\frac{8s-2N}{N}}-N^2|\phi_p|_{2}^{\frac{8s}{N}}} 处取得极大值, 且无极小值点；对于 c\leq c_3 , f_p(t) 无极小值点. (i) 当 0<p<\frac{4s}{N} 时, 即 \frac{NP}{4s}<1 , 易知函数 f_p(t) 有一个唯一的极小点, 设为 t_1>0 . 因此, 对 \forall\ c>0 , f_p(t)\geq f_p(t_1) . (ii) 当 p=\frac{4s}{N} 时, 通过计算知 \alpha_p\beta_p=\frac{1}{2} , 且 f_p(t)=\frac{b}{4}t^2+\frac{1}{2}(a-\frac{K(4s+2N)c^{\frac{4s}{N}}}{N|\phi_p|_{2}^{\frac{4s}{N}}})t-Kc^2 . a-\frac{K(4s+2N)c^{\frac{4s}{N}}}{N|\phi_p|_{2}^{\frac{4s}{N}}}<0 , 即 c>c_1=(\frac{aN}{K(4s+2N)})^{\frac{N}{4s}}|\phi_p|_{2} 时, 由 f'_p(t)=0 , 计算可得极小点为 t_2=\frac{K(4s+2N)c^{\frac{4s}{N}}}{bN|\phi_p|_{2}^{\frac{4s}{N}}}-\frac{a}{b} , 且 而当 c\leq c_1=(\frac{aN}{K(4s+2N)})^{\frac{N}{4s}}|\phi_p|_{2} 时, 因为 f_p(t)$$ (0, +\infty)$上为增函数, 即$f_p(t)\geq -Kc^2$, 因此当$t>0$时, 函数$f_p(t)$无极小值点.

(iii) 当$\frac{4s}{N}<p<\frac{8s}{N}$时, 有$f_p(t)=\frac{b}{4}t^2+\frac{a}{2}t-\frac{K(p+2)c^{p+2-\frac{Np}{2s}}}{|\phi_p|_{2}^p}\alpha_p\beta_pt^{\frac{NP}{4s}}-Kc^2$, 其中$1<\frac{NP}{4s}<2$, 取$\gamma=\frac{8s-Np}{4s} $$\xi=1-\gamma=\frac{Np-4s}{4s} , 由Young不等式, 对任意的 t>0 , 有 上式中第二个 "=" 成立当且仅当 \frac{a}{2\gamma}t=\frac{b}{4\xi}t^2 , 由此即得 t\triangleq t_3=:\frac{2a\xi}{b\gamma}=\frac{2(Np-4s)a}{(8s-Np)b} . 利用上面的不等式可得 \begin{eqnarray} f_p(t)&\geq &\left(\left(\frac{2as}{8s-Np}\right)^{\frac{8s-Np}{4s}}\left(\frac{bs}{Np-4s}\right)^{\frac{Np-4s}{4s}}-\frac{K\alpha_p\beta_p(p+2)c^{p+2-\frac{NP}{2s}}}{|\phi_p|_{2}^p}\right){t_3}^{\frac{Np}{4s}}-Kc^2 \\ &=&f_p(t_3). \end{eqnarray} 此时, 令 c<c_2 时, 恒有 f_p(t)> -Kc^2 , 此时 f_p(t) 无极小值点；当 c\geq c_2 时, f_p(t)$$ t_3=\frac{2(Np-4s)a}{(8s-Np)b}$处达到极小值

(iv) 当$p=\frac{8s}{N}, N < 4s$时, 则$f_p(t)=\frac{a}{2}t+\frac{1}{4}(b-\frac{K(4s+N)(4s-N)c^{\frac{8s-2N}{N}}}{N^2|\phi_p|_{2}^{\frac{8s}{N}}})t^2-Kc^2$, 再根据$f'_p(t)=\frac{a}{2}+\frac{1}{2}(b-\frac{K(4s+N)(4s-N)c^{\frac{8s-2N}{N}}}{N^2|\phi_p|_{2}^{\frac{8s}{N}}})t=0$, 得$t_4=\frac{N^2|\phi_p|_{2}^{\frac{8s}{N}}}{K(4s+N)(4s-N)c^{\frac{8s-2N}{N}}-bN^2|\phi_p|_{2}^{\frac{8s}{N}}}$.

由文献[3] 可知$\| u\| _{H^s}$范数具有弱下半连续性, 即

$$$\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u|^2{\rm d}x\leq \bigg(\liminf\limits_{n\rightarrow \infty} \bigg(\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u_n|^2{\rm d}x\bigg)^{\frac{1}{2}}\bigg)^2 \leq \liminf\limits_{n\rightarrow \infty} \int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u_n|^2{\rm d}x,$$$

$$$\bigg(\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u|^2{\rm d}x\bigg)^2\leq \liminf\limits_{n\rightarrow \infty} \bigg(\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u_n|^2{\rm d}x\bigg)^2.$$$

$u_n(x)\leq \frac{c_1N^{\frac{1}{2}}}{|S^{N-1}|^{\frac{1}{2}}|x|^{\frac{N}{2}}}\leq \frac{c_1N^{\frac{1}{2}}}{|S^{N-1}|^{\frac{1}{2}}R^{\frac{N}{2}}}$, 对$\forall |x|\geq R$, 这意味着在此情形满足条件$(G_{3})$. 因此对任意$\varepsilon >0$, $R$足够大, 由条件$(G_{3})$

$$$\lim\limits_{n\rightarrow \infty}\int_{{{\Bbb R}} ^N} G(|x|, u_n){\rm d}x=\int_{{{\Bbb R}} ^N} G(|x|, u){\rm d}x,$$$

## 3 定理证明

首先由条件$(G_{1}) $$(G_{2}) \begin{eqnarray} \int_{{{\Bbb R}} ^N}G(|x|, u(x)){\rm d}x&\leq &\int_{{{\Bbb R}} ^N}G(|x|, |u(x)|){\rm d}x \leq \int_{{{\Bbb R}} ^N}K(|u|^2+|u|^{p+2}){\rm d}x \\ &=&Kc^2+K\int_{{{\Bbb R}} ^N}|u|^{p+2}{\rm d}x. \end{eqnarray} t=\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2}u|^2{\rm d}x , 利用分数阶Gagliardo-Nirenberg不等式(1.3), 可得 \begin{eqnarray} E(u)&\geq &\frac{a}{2}\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2} u|^2{\rm d}x+\frac{b}{4}\bigg(\int_{{{\Bbb R}} ^N} |(-\Delta )^\frac{s}{2} u|^2{\rm d}x\bigg)^2-Kc^2 \\ &&-\frac{K(p+2)c^{p+2-\frac{Np}{2s}}}{|\phi_p|_{2}^{p}}\alpha_p\beta_p \left(\int_{{{\Bbb R}} ^N}|(-\Delta )^\frac{s}{2}u|^2{\rm d}x\right)^{\frac{Np}{4s}} \\ &\triangleq& f_p(t). \end{eqnarray} (i) 当 0<p<\frac{4s}{N} 时, 由引理2.1中(i) 知, 对 \forall\ c>0 , f_p(t)\geq f_p(t_1) , 因此 m(c)\geq f_p(t)\geq f_p(t_1) , 即 m(c)> -\infty . \{u_n\} 是极小化序列且满足 \int_{{{\Bbb R}} ^N}|u_n|^2{\rm d}x=c^2 , 由 m(c) 的定义, 对 \forall\ \varepsilon >0 , 有 E(u_n)<m(c)+\varepsilon , 结合不等式 (3.2) 可知存在常数 M>0 , 使得 \int_{{{\Bbb R}} ^N}|(-\Delta )^\frac{s}{2}u_n|^2{\rm d}x<M , 进一步可得 \| u_n\| _{H^s({{\Bbb R}} ^N)}=\left(\int_{{{\Bbb R}} ^N} \left(|(-\Delta )^\frac{s}{2}u_n|^2+|u_n|^2\right){\rm d}x\right)^{\frac{1}{2}} 有界, 即极小化序列 \{u_n\}$$ H^{s}({{\Bbb R}} ^{N})$中有界. 综上可知, 存在有界序列$\{u_n\}\subset H^{s}({{\Bbb R}} ^{N})$使得${ } \lim_{n\rightarrow +\infty}E(u_n)=m(c)=\inf \left\{E(\omega):\omega \in S_c\right\}$.$\{u_n\}\subset H^{s}({{\Bbb R}} ^{N}) $$|u_n|\subset L^{2}({{\Bbb R}} ^{N}) |u_n|\in H^s({{\Bbb R}} ^N) . \|u_n\|_{2}^2 =\|| u_n|\|_{2}^2=c^2 , 进而 |u_n| \in S_c . 由条件 (G_{1}) 现设 u_n^*$$ |u_n|$的Schwarz对称递减重排. 因为$u_n\in H^s({{\Bbb R}} ^N)$, 由文献[26], 知$u_n^* \in H^s({{\Bbb R}} ^N)$, 且$\|u_n^*\|_{2}^2 =\|u_n\|_{2}^2=c^2$, 因此, $u_n^* \in S_c$. 再利用文献[27] 和条件$(G_{1})$, 可得

$E(v)\leq \tilde{m}_c$.$\| v\| _2^2=:d^2\leq c^2$, 不妨假设$0<d<c$, 则$\tilde{m}_c<\tilde{m}_d$, 且

$u_{\lambda}(x) \in S_c$.

$p= \frac{8s}{N}$时, 由$\liminf\limits_{s\rightarrow 0}\frac{G(r, s)}{s^{p+4}}=A>0$, 对$\forall\ \varepsilon>0$, $\exists \ \delta>0$, 当$|s|< \delta$时, $G(r, s)>(A-\varepsilon)|s|^{p+4}$, 因此当$p= \frac{8s}{N}$, 即$\frac{Np}{2}=4s$时, 可得

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Amick C J , Toland J F .

Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane

Acta Mathematica, 1991, 167 (1): 107- 126

Chen W X , Li C M , Ou B .

Classification of solutions for an integral equation

Communications on Pure and Applied Mathematics, 2006, 59 (3): 330- 343

Nezza E Di , Palatucci G , Valdinoci E .

Hitchhiker's guide to the fractional Sobolev spaces

Bull Sci Math, 2012, 136 (5): 521- 573

Ou B , Li C , Chen W .

Qualitative properties of solutions for an integral equation

Discrete and Continuous Dynamical Systems-Series A, 2017, 12 (2): 347- 354

Li Y Y .

Remark on some conformally invariant integral equations: the method of moving spheres

Journal of the European Mathematical Society, 2004, 6 (2): 153- 180

Frank R L , Lenzmann E .

Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$

Acta Mathematica, 2013, 210 (2): 261- 318

Frank R L , Lenzmann E , Silvestre L .

Uniqueness of radial solutions for the fractional Laplacian

Communications on Pure and Applied Mathematics, 2013, 69 (9): 1671- 1726

Cheng Y Q , Teng K M .

Positive ground state solutions for nonlinear critical Kirchhoff type problem

Acta Mathematica Scientia, 2021, 41A (3): 666- 685

Deng Y B , Peng S J , Shuai W .

Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$

J Funct Anal, 2015, 269 (11): 3500- 3527

Guo H L , Wang Y B .

A remark on a constrained variational problem

Acta Mathematica Scientia, 2017, 37A (6): 1125- 1128

Ye H Y .

The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations

Math Methods Appl Sci, 2015, 38 (13): 2663- 2679

DOI:10.1002/mma.3247

Liu Z D , Wang Z P .

Least energy solution for nonlinear Kirchhoff type elliptic equation

Acta Mathematica Scientia, 2019, 39A (2): 264- 276

$\mathbb{R}^{N}$上的Kirchhoff型问题非平凡解的存在性和多解性

Wei M C , Tang C L .

Existence and multiplicity of nontrivial solutions for Kirchhoff-type problem in begin{document}$mathbb{R}^{N}$end{document}

Acta Mathematica Scientia, 2015, 35A (1): 151- 162

Kirchhoff型方程有关的非线性方程多解的存在性

Liang W C , Zhang Z J .

Multiple solutions for nonlinear equations related to Kirchhoff type equations

Acta Mathematica Scientia, 2020, 40A (4): 842- 849

Mao A M , Zhang Z T .

Sign-changing and multiple solutions of Kirchhoff type problems without P.S. condition

Nonlinear Anal, 2009, 70 (3): 1275- 1287

He X M , Zou W M .

Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$

J Differ Equ, 2012, 252 (2): 1813- 1834

Li G B , Niu Y H .

The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation

Acta Mathematica Scientia, 2020, 40B (1): 90- 112

Kirchhoff方程单峰解的局部唯一性

Xu S M , Wang C H .

Local uniqueness of a single peak solution of a subcritical Kirchhoff problem in begin{document}$\mathbb{R}^{3}$end{document}

Acta Mathematica Scientia, 2020, 40A (2): 432- 440

Zeng X Y , Zhang Y M .

Existence and uniqueness of normalized solutions for the Kirchhoff equation

Applied Math Letters, 2017, 74, 52- 59

Guo H L , Zhang Y M , Zhou H S .

Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential

Commun Pure Appl Aanl, 2018, 17 (5): 1875- 1897

Li R X , Wang W Q , Zeng X Y .

A constrained variational problem of Kirchhoff type equation with ellipsoid-shaped potential

Acta Mathematica Scientia, 2019, 39A (6): 1323- 1333

Liu Z S, Squassina M, Zhang J J. Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. Nonlinear Differential Equations and Applications, 2017, 24: Article number 50

Cheng K , Gao Q .

Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$

Acta Mathematica Scientia, 2018, 38B (6): 1712- 1730

Pucci P , Xiang M Q , Zhang B L .

Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$

Calc Var Partial Differ Equ, 2015, 54, 2785- 2806

Huang X M , Zhang Y M .

Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation

Mathematical Methods in the Applied Sciences, 2020, 43 (15): 8763- 8775

Frank R L , Seiringer R .

Non-linear ground state representations and sharp Hardy inequalities

J Fun Anal, 2008, 255 (12): 3407- 3430

Burchard A , Hajaiej H .

Rearrangement inequalities for functionals with monotone integrands

Journal of Functional Analysis, 2006, 233 (2): 561- 582

/

 〈 〉