# American Institute of Mathematical Sciences

doi: 10.3934/era.2021038
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## 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:
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##### References:
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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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