    doi: 10.3934/era.2021038

## Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

 School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, P.R. China

* Corresponding author: Chungen Liu

Received  March 2021 Revised  April 2021 Early access  May 2021

Fund Project: The first author is Partially supported by the NSF of China (11790271), Guangdong Basic and Applied basic Research Foundation(2020A1515011019), Innovation and Development Project of Guangzhou University

In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem
 $\begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&x\in\Omega,\\ u = 0,&x\in\mathbb{R}^{3}\backslash\Omega, \end{cases}$
where
 $k$
is a positive parameter,
 $\mathcal{L}_K$
stands for a nonlocal fractional operator which is defined with the kernel function
 $K$
. By using the nodal Nehari manifold method, we obtain a least energy nodal solution
 $u$
and a ground state solution
 $v$
to this problem when
 $k\gg1$
, where the nonlinear function
 $f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R}$
is a Carathéodory function.
Citation: Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, doi: 10.3934/era.2021038
##### References:
  N. Abatangelo and X. Ros-Oton, Obstacle problems for integro-differential operators: Higher regularity of free boundaries, Advances in Mathematics, 360 (2020), 106931. doi: 10.1016/j.aim.2019.106931.  Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar  K. Cheng and Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$, Acta Mathematica Scientia Ser. B (Engl. Ed.), 38 (2018), 1712-1730.  doi: 10.1016/S0252-9602(18)30841-5.  Google Scholar  G. Gu, Y. Yu and F. Zhao, The least energy sign-changing solution for a nonlocal problem, J. Math. Phys., 58 (2017), 051505. doi: 10.1063/1.4982960.  Google Scholar  G. Gu, W. Zhang and F. Zhao, Infinitely many sign-changing solutions for a nonlocal problem, Annali di Matematica Pura Appl., 197 (2018), 1429-1444.  doi: 10.1007/s10231-018-0731-2.  Google Scholar  G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar  J. Korvenpää, T. Kuusi and G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differ. Equ., 55 (2016), 63. doi: 10.1007/s00526-016-0999-2.  Google Scholar  T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar  H. Lu and X. Zhang, Positive solution for a class of nonlocal elliptic equations, Applied Mathematics Letters, 88 (2019), 125-131.  doi: 10.1016/j.aml.2018.08.019.  Google Scholar  H. Luo, X. Tang and Z. Gao, Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains, J. Math. Phys., 59 (2018), 031504. doi: 10.1063/1.5026674.  Google Scholar  Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.  doi: 10.1007/BF02559588.  Google Scholar  R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar  R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar  W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259 (2015), 1256-1274.  doi: 10.1016/j.jde.2015.02.040.  Google Scholar  X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.  Google Scholar  K. Teng, Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators, Annali di Matematica Pura ed Applicata, 194 (2015), 1455-1468.  doi: 10.1007/s10231-014-0428-0.  Google Scholar  D.-B. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501, 19 pp.. doi: 10.1063/1.5074163.  Google Scholar  T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3.  Google Scholar  Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar  J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J Math Anal Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar

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##### References:
  N. Abatangelo and X. Ros-Oton, Obstacle problems for integro-differential operators: Higher regularity of free boundaries, Advances in Mathematics, 360 (2020), 106931. doi: 10.1016/j.aim.2019.106931.  Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar  K. Cheng and Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$, Acta Mathematica Scientia Ser. B (Engl. Ed.), 38 (2018), 1712-1730.  doi: 10.1016/S0252-9602(18)30841-5.  Google Scholar  G. Gu, Y. Yu and F. Zhao, The least energy sign-changing solution for a nonlocal problem, J. Math. Phys., 58 (2017), 051505. doi: 10.1063/1.4982960.  Google Scholar  G. Gu, W. Zhang and F. Zhao, Infinitely many sign-changing solutions for a nonlocal problem, Annali di Matematica Pura Appl., 197 (2018), 1429-1444.  doi: 10.1007/s10231-018-0731-2.  Google Scholar  G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar  J. Korvenpää, T. Kuusi and G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differ. Equ., 55 (2016), 63. doi: 10.1007/s00526-016-0999-2.  Google Scholar  T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar  H. Lu and X. Zhang, Positive solution for a class of nonlocal elliptic equations, Applied Mathematics Letters, 88 (2019), 125-131.  doi: 10.1016/j.aml.2018.08.019.  Google Scholar  H. Luo, X. Tang and Z. Gao, Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains, J. Math. Phys., 59 (2018), 031504. doi: 10.1063/1.5026674.  Google Scholar  Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.  doi: 10.1007/BF02559588.  Google Scholar  R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar  R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar  W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259 (2015), 1256-1274.  doi: 10.1016/j.jde.2015.02.040.  Google Scholar  X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.  Google Scholar  K. Teng, Two nontrivial solutions for an elliptic problem involving some nonlocal integro-differential operators, Annali di Matematica Pura ed Applicata, 194 (2015), 1455-1468.  doi: 10.1007/s10231-014-0428-0.  Google Scholar  D.-B. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501, 19 pp.. doi: 10.1063/1.5074163.  Google Scholar  T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3.  Google Scholar  Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463.  doi: 10.1016/j.jmaa.2005.06.102.  Google Scholar  J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J Math Anal Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032.  Google Scholar
  Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857  Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289  Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292  Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030  Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021108  Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108  Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285  Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126  Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143  Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326  Yanjun Liu, Chungen Liu. Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2819-2838. doi: 10.3934/cpaa.2020123  Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214  Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257  Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943  Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895  Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499  Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091  Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064  Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173  Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021026

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