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April  2020, 40(4): 2189-2212. doi: 10.3934/dcds.2020111

Semilinear elliptic system with boundary singularity

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  May 2019 Revised  October 2019 Published  January 2020

Fund Project: The authors are supported in part by the National Natural Science Foundation of China (11631002 and 11871102).

In this paper, we investigate the asymptotic behavior of local solutions for the semilinear elliptic system $ -\Delta \mathbf{u} = |\mathbf{u}|^{p-1}\mathbf{u} $ with boundary isolated singularity, where $ 1<p<\frac{n+2}{n-2} $, $ n\geq 2 $ and $ \mathbf{u} $ is a $ C^2 $ nonnegative vector-valued function defined on the half space. This work generalizes the correspondence results of Bidaut-Véron-Ponce-Véron on the scalar case, and Ghergu-Kim-Shahgholian on the internal singularity case.

Citation: Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111
References:
[1]

M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302. Cambridge University Press, Cambridge, 2004.  Google Scholar

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. PureAppl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

P. Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.  doi: 10.1007/BF01210610.  Google Scholar

[4]

M. F. Bidaut-VéronA. C. Ponce and L. Véron, Isolated boundary singularities of semilinear elliptic equations, Calc. Var. Partial Differential Equations, 40 (2011), 183-221.  doi: 10.1007/s00526-010-0337-z.  Google Scholar

[5]

M. F. Bidaut-VéronA. C. Ponce and L. Véron, Boundary singularities of positive solutions of some nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris, 344 (2007), 83-88.  doi: 10.1016/j.crma.2006.11.027.  Google Scholar

[6]

M. F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: The subcritical case, Rev. Mat. Iberoam., 16 (2000), 477-513.  doi: 10.4171/RMI/281.  Google Scholar

[7]

H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614.  doi: 10.1080/03605307708820041.  Google Scholar

[8]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[9]

R. CajuJ. M. do Ó and A. Silva Santos, Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1575-1601.  doi: 10.1016/j.anihpc.2019.02.001.  Google Scholar

[10]

Z. J. ChenC.-S. Lin and W. M. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.  doi: 10.1016/j.jfa.2013.08.021.  Google Scholar

[11]

E. N. Dancer, Some notes on the method of moving planes, Bull. Aust. Math. Soc., 46 (1992), 425-434.  doi: 10.1017/S0004972700012089.  Google Scholar

[12]

M. del PinoM. Musso and F. Pacard, Boundary singularities for weak solutions of semilinear elliptic problems, J. Funct. Anal., 253 (2007), 241-272.  doi: 10.1016/j.jfa.2007.05.023.  Google Scholar

[13]

O. DruetE. Hebey and J. Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059.  doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[14]

M. Ghergu, S. Kim and H. Shahgholian, Isolated Singularities for Semilinear Elliptic Systems with Power-Law Nonlinearity, arXiv: 1804.04291. Google Scholar

[15]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[16]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[17]

A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J., 60 (1991), 271-324.  doi: 10.1215/S0012-7094-91-06414-8.  Google Scholar

[18]

Y. X. Guo, Non-existence, monotonicity for positive solutions of semilinear elliptic system in $ \mathbb{R}^n_+$, Commun. Contemp. Math., 12 (2010), 351-372.  doi: 10.1142/S0219199710003853.  Google Scholar

[19]

Z. C. HanY. Y. Li and E. V. Teixeira, Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities, Invent. Math., 182 (2010), 635-684.  doi: 10.1007/s00222-010-0274-7.  Google Scholar

[20]

N. KorevaarR. MazzeoF. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.  doi: 10.1007/s002220050285.  Google Scholar

[21]

Y. Y. Li, Conformally invariant fully nonlinear elliptic equations and isolated singularities, J. Funct. Anal., 233 (2006), 380-425.  doi: 10.1016/j.jfa.2005.08.009.  Google Scholar

[22]

P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations, 38 (1980), 441-450.  doi: 10.1016/0022-0396(80)90018-2.  Google Scholar

[23]

P. PoláčikP. 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

[24]

A. Porretta and L. Véron, Separable solutions of quasilinear Lane-Emden equations, J. Eur. Math. Soc., 15 (2011), 755-774.  doi: 10.4171/JEMS/375.  Google Scholar

[25]

J. G. Xiong, The critical semilinear elliptic equation with boundary isolated singularities, J. Differential Equations, 263 (2017), 1907-1930.  doi: 10.1016/j.jde.2017.03.034.  Google Scholar

show all references

References:
[1]

M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302. Cambridge University Press, Cambridge, 2004.  Google Scholar

[2]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. PureAppl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[3]

P. Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.  doi: 10.1007/BF01210610.  Google Scholar

[4]

M. F. Bidaut-VéronA. C. Ponce and L. Véron, Isolated boundary singularities of semilinear elliptic equations, Calc. Var. Partial Differential Equations, 40 (2011), 183-221.  doi: 10.1007/s00526-010-0337-z.  Google Scholar

[5]

M. F. Bidaut-VéronA. C. Ponce and L. Véron, Boundary singularities of positive solutions of some nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris, 344 (2007), 83-88.  doi: 10.1016/j.crma.2006.11.027.  Google Scholar

[6]

M. F. Bidaut-Véron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: The subcritical case, Rev. Mat. Iberoam., 16 (2000), 477-513.  doi: 10.4171/RMI/281.  Google Scholar

[7]

H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614.  doi: 10.1080/03605307708820041.  Google Scholar

[8]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[9]

R. CajuJ. M. do Ó and A. Silva Santos, Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1575-1601.  doi: 10.1016/j.anihpc.2019.02.001.  Google Scholar

[10]

Z. J. ChenC.-S. Lin and W. M. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.  doi: 10.1016/j.jfa.2013.08.021.  Google Scholar

[11]

E. N. Dancer, Some notes on the method of moving planes, Bull. Aust. Math. Soc., 46 (1992), 425-434.  doi: 10.1017/S0004972700012089.  Google Scholar

[12]

M. del PinoM. Musso and F. Pacard, Boundary singularities for weak solutions of semilinear elliptic problems, J. Funct. Anal., 253 (2007), 241-272.  doi: 10.1016/j.jfa.2007.05.023.  Google Scholar

[13]

O. DruetE. Hebey and J. Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059.  doi: 10.1016/j.jfa.2009.07.004.  Google Scholar

[14]

M. Ghergu, S. Kim and H. Shahgholian, Isolated Singularities for Semilinear Elliptic Systems with Power-Law Nonlinearity, arXiv: 1804.04291. Google Scholar

[15]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[16]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[17]

A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J., 60 (1991), 271-324.  doi: 10.1215/S0012-7094-91-06414-8.  Google Scholar

[18]

Y. X. Guo, Non-existence, monotonicity for positive solutions of semilinear elliptic system in $ \mathbb{R}^n_+$, Commun. Contemp. Math., 12 (2010), 351-372.  doi: 10.1142/S0219199710003853.  Google Scholar

[19]

Z. C. HanY. Y. Li and E. V. Teixeira, Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities, Invent. Math., 182 (2010), 635-684.  doi: 10.1007/s00222-010-0274-7.  Google Scholar

[20]

N. KorevaarR. MazzeoF. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.  doi: 10.1007/s002220050285.  Google Scholar

[21]

Y. Y. Li, Conformally invariant fully nonlinear elliptic equations and isolated singularities, J. Funct. Anal., 233 (2006), 380-425.  doi: 10.1016/j.jfa.2005.08.009.  Google Scholar

[22]

P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations, 38 (1980), 441-450.  doi: 10.1016/0022-0396(80)90018-2.  Google Scholar

[23]

P. PoláčikP. 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

[24]

A. Porretta and L. Véron, Separable solutions of quasilinear Lane-Emden equations, J. Eur. Math. Soc., 15 (2011), 755-774.  doi: 10.4171/JEMS/375.  Google Scholar

[25]

J. G. Xiong, The critical semilinear elliptic equation with boundary isolated singularities, J. Differential Equations, 263 (2017), 1907-1930.  doi: 10.1016/j.jde.2017.03.034.  Google Scholar

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