American Institute of Mathematical Sciences

April  2021, 20(4): 1681-1698. doi: 10.3934/cpaa.2021036

The regularity lifting methods for nonnegative solutions of Lane-Emden system

 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  June 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author is partially supported by NSFC-12031012 and NSFC-11831003

In this paper, we focus on the regularity of nonnegative solutions of Lane-Emden system
 $\begin{equation*} \begin{cases} -\Delta u = v^p\\ -\Delta v = u^q \end{cases} \mbox{ in } \mathbb{R}^n. \end{equation*}$
By means of Kelvin transform, we turn this problem into estimating the local integrability of
 $(\bar{u},\bar{v})$
. Assume that
 $(\bar{u},\bar{v})$
possesses some initial local integrability beforehand.
 $(\bar{u},\bar{v})\in L_{loc}^{r_0}(\mathbb{R}^n)\times L_{loc}^{s_0}(\mathbb{R}^n)$
for any suitable
 $r_0$
and
 $s_0$
under specified conditions. Then through a regularity lifting method by contracting operators, we prove that
 $(\bar{u},\bar{v})\in L_{loc}^r(\mathbb{R}^n)\times L_{loc}^s(\mathbb{R}^n)$
for
 $r$
and
 $s$
sufficiently large under twice regularity lifting if needed. Furthermore, we lift the regularity of solutions to
 $L^\infty(\mathbb{R}^n)\times L^\infty(\mathbb{R}^n).$
We believe that these new methods employed in this paper can be widely applied to study a variety of other problems with different spaces and linear or nonlinear problems.
Citation: Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036
References:
 [1] W. Chen and C. Li, An integral system and the LaneEmden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [2] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010.  Google Scholar [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar [4] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.  Google Scholar [5] Z. Cheng and G. Huang, A Liouville theorem for the subcritical Lane-Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.  doi: 10.3934/dcds.2019058.  Google Scholar [6] L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010. Google Scholar [7] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.   Google Scholar [8] C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proc. Natl. Acad. Sci., 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.  Google Scholar [9] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [10] L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [11] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479.   Google Scholar [12] E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar [13] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar [14] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [15] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar [16] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380.   Google Scholar [17] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar [18] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.  Google Scholar [19] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar [20] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [21] M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258.   Google Scholar

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References:
 [1] W. Chen and C. Li, An integral system and the LaneEmden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [2] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010.  Google Scholar [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar [4] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.  Google Scholar [5] Z. Cheng and G. Huang, A Liouville theorem for the subcritical Lane-Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.  doi: 10.3934/dcds.2019058.  Google Scholar [6] L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010. Google Scholar [7] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.   Google Scholar [8] C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proc. Natl. Acad. Sci., 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.  Google Scholar [9] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar [10] L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar [11] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479.   Google Scholar [12] E. Mitidieri, A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar [13] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar [14] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [15] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar [16] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380.   Google Scholar [17] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar [18] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.  Google Scholar [19] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar [20] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [21] M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258.   Google Scholar
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