# American Institute of Mathematical Sciences

June  2021, 14(6): 1995-2023. doi: 10.3934/dcdss.2021042

## Positive least energy solutions for k-coupled critical systems involving fractional Laplacian

 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

* Corresponding author: Wenming Zou

Received  December 2020 Revised  February 2021 Published  June 2021 Early access  April 2021

Fund Project: The second author is supported by NSFC(11771234, 11871123)

In this paper, we study the following
 $k$
-coupled critical system:
 \begin{equation*} \left\{ \begin{aligned} (-\Delta)^s u_i +\lambda_iu_i & = \mu_i u_i^{2^*-1}+\sum\limits_{j = 1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega,\\ u_i & = 0 \quad \hbox{in}\;\; \; \mathbb R^N\backslash\Omega, \quad i = 1,2,\cdots, k. \end{aligned} \right. \end{equation*}
Here
 $(-\Delta)^s$
is the fractional Laplacian operator,
 $0 , $ 2^{*} = \frac{2N}{N-2s} $is a fractional Sobolev critical exponent, $ N>2s $, $ - \lambda_s( \Omega)< \lambda_i<0, \mu_i>0 $, $ \beta_{ij} = \beta_{ji}\ne 0 $and $ \Omega\subset {\mathbb R}^N $is a smooth bounded domain, where $ \lambda_s( \Omega) $is the first eigenvalue of $ (-\Delta)^{s} $with the homogeneous Dirichlet boundary datum. We characterize the positive least energy solution of the $ k $-coupled fractional critical system for the purely cooperative case $ \beta_{ij}>0 $with $ N> 4s $. We shall introduce the idea of induction to prove our results. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $ k $-coupled system decreases as $ k $grows. Moreover, we establish the existence of positive least energy solution of the limit system in $ \mathbb R^N $, as well as classification results. Meanwhile, we also construct a positive solution for a more general system involving subcritical items. Besides, we investigated in the asymptotic behaviour of the positive least energy solutions of the critical system. We point out that the results of the fractional critical systems have some coincidences with those of the critical Schrödinger systems. Citation: Xin Yin, Wenming Zou. Positive least energy solutions for k-coupled critical systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1995-2023. doi: 10.3934/dcdss.2021042 ##### References:  [1] B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole$ \mathbb R^N$, Calc. Var. PDE, 34 (2009), 97-137. doi: 10.1007/s00526-008-0177-2. [2] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Equ., 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [3] G. Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016. doi: 10.1017/CBO9781316282397. [4] W. Chen and S. Deng, Multiple solutions for a critical fractional elliptic system involving concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1167-1193. [5] Z. Chen and C. Lin, Asymptotic behavior of least energy solutions for a critical elliptic system, Int. Math. Res. Not., 21 (2015), 11045-11082. doi: 10.1093/imrn/rnv016. [6] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8. [7] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. PDE., 52 (2015), 423-467. doi: 10.1007/s00526-014-0717-x. [8] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. [9] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian,, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. [10] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities,, Comm. Pure Appl. Anal., 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285. [11] C. Mou, Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space, Comm. Pure Appl. Anal., 14 (2015), 2335-2362. doi: 10.3934/cpaa.2015.14.2335. [12] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [13] Y. Park, Fractional Polya–Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271. [14] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [15] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628. doi: 10.1007/s00205-014-0740-2. [16] 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. [17] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. [18] R. Servadei and E. Valdinoci, The Brézis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. [19] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [20] Y. Wu, Ground states of a K-component critical system with linear and nonlinear couplings: the attractive case, Adv. Nonlinear Stud., 19 (2019), 595-623. doi: 10.1515/ans-2019-2049. [21] X. Yin and W. Zou, Positive least energy solutions for k-coupled Schrödinger system with critical exponent: The higher dimension and cooperative case, submitted. [22] X. Yu, Liouville type theorems for integral equations and integral systems, Cal. Var. PDE., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. [23] M. Zhen, J. He and H. Xu, Critical system involving fractional Laplacian, Comm. Pure. Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013. show all references ##### References:  [1] B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole$ \mathbb R^N$, Calc. Var. PDE, 34 (2009), 97-137. doi: 10.1007/s00526-008-0177-2. [2] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Equ., 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [3] G. Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 2016. doi: 10.1017/CBO9781316282397. [4] W. Chen and S. Deng, Multiple solutions for a critical fractional elliptic system involving concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1167-1193. [5] Z. Chen and C. Lin, Asymptotic behavior of least energy solutions for a critical elliptic system, Int. Math. Res. Not., 21 (2015), 11045-11082. doi: 10.1093/imrn/rnv016. [6] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551. doi: 10.1007/s00205-012-0513-8. [7] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. PDE., 52 (2015), 423-467. doi: 10.1007/s00526-014-0717-x. [8] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. [9] Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian,, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. [10] X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities,, Comm. Pure Appl. Anal., 15 (2016), 1285-1308. doi: 10.3934/cpaa.2016.15.1285. [11] C. Mou, Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space, Comm. Pure Appl. Anal., 14 (2015), 2335-2362. doi: 10.3934/cpaa.2015.14.2335. [12] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [13] Y. Park, Fractional Polya–Szegö inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271. [14] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [15] X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628. doi: 10.1007/s00205-014-0740-2. [16] 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. [17] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. [18] R. Servadei and E. Valdinoci, The Brézis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4. [19] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [20] Y. Wu, Ground states of a K-component critical system with linear and nonlinear couplings: the attractive case, Adv. Nonlinear Stud., 19 (2019), 595-623. doi: 10.1515/ans-2019-2049. [21] X. Yin and W. Zou, Positive least energy solutions for k-coupled Schrödinger system with critical exponent: The higher dimension and cooperative case, submitted. [22] X. Yu, Liouville type theorems for integral equations and integral systems, Cal. Var. PDE., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z. [23] M. Zhen, J. He and H. Xu, Critical system involving fractional Laplacian, Comm. Pure. Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013.  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