American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls
  • DCDS Home
  • This Issue
  • Next Article
    Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions
April  2020, 40(4): 2189-2212. doi: 10.3934/dcds.2020111

Semilinear elliptic system with boundary singularity

Yimei Li and  Jiguang Bao

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.

Keywords: Elliptic systems, boundary isolated singularity, asymptotic analysis.
Mathematics Subject Classification: 35J61, 35B40, 35B53.
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. Agmon, A. 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-Véron, A. 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-Véron, A. 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. Caffarelli, B. 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. Caju, J. 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. Chen, C.-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 Pino, M. 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. Druet, E. 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. Gidas, W. 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. Han, Y. 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. Korevaar, R. Mazzeo, F. 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. 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

[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. Agmon, A. 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-Véron, A. 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-Véron, A. 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. Caffarelli, B. 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. Caju, J. 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. Chen, C.-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 Pino, M. 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. Druet, E. 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. Gidas, W. 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. Han, Y. 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. Korevaar, R. Mazzeo, F. 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. 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

[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

[1]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
[2]

Jianfeng Huang, Yulin Zhao. Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2861-2883. doi: 10.3934/dcds.2013.33.2861
[3]

Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376
[4]

Monica De Angelis, Pasquale Renno. Asymptotic effects of boundary perturbations in excitable systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2039-2045. doi: 10.3934/dcdsb.2014.19.2039
[5]

Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53
[6]

Soumya Kundu, Soumitro Banerjee, Damian Giaouris. Vanishing singularity in hard impacting systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 319-332. doi: 10.3934/dcdsb.2011.16.319
[7]

Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126
[8]

Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453
[9]

Mathias Schäffner, Anja Schlömerkemper. On Lennard-Jones systems with finite range interactions and their asymptotic analysis. Networks & Heterogeneous Media, 2018, 13 (1) : 95-118. doi: 10.3934/nhm.2018005
[10]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[11]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
[12]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[13]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[14]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[15]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[16]

Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072
[17]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[18]

Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure & Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531
[19]

Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549
[20]

Moncef Aouadi, Taoufik Moulahi. Asymptotic analysis of a nonsimple thermoelastic rod. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1475-1492. doi: 10.3934/dcdss.2016059

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (131)
  • HTML views (104)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences