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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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

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