American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces
  • DCDS Home
  • This Issue
  • Next Article
    On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model
November  2014, 34(11): 4947-4966. doi: 10.3934/dcds.2014.34.4947

Liouville type theorem for nonlinear elliptic equation with general nonlinearity

Xiaohui Yu 1,

1. 

The Center for China's Overseas Interests, Shenzhen University, Shenzhen Guangdong, 518060, China

Received  September 2013 Revised  December 2013 Published  May 2014

In this paper, we study the nonexistence of positive solutions for the following elliptic equation $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) & in \quad \mathbb{R}_+^N, \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) & on \quad \partial \mathbb{R}_+^N \end{array} \right. $$ and elliptic system $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u_1=f_1(u_1,u_2) &in \quad \mathbb{R}_+^N, \\ \\-\Delta u_2=f_2(u_1,u_2) & in\quad \mathbb{R}_+^N, \\ \displaystyle \\ \frac{\partial u_1}{\partial \nu}=g_1(u_1,u_2),\quad \frac{\partial u_2}{\partial \nu}=g_2(u_1,u_2) & on \quad \partial \mathbb{R}_+^N. \end{array} \right. $$ We will prove that these problems possess no positive solutions under some assumptions on nonlinear terms. The main technique we use is the moving plane method in an integral form.
Keywords: moving planes, Liouville type theorem, elliptic equation., maximum principle.
Mathematics Subject Classification: Primary: 35J60, 35J57; Secondary: 35J1.
Citation: Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
References:
[1]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[2]

Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[3]

Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[4]

AIMS Book Series, vol. 4, 2010.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[6]

Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[7]

Comm. P.D.E., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[8]

J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[9]

Rev. Mat. Iberoamericana, 20 (2004), 67-86.  Google Scholar

[10]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.  Google Scholar

[11]

J. Math. Pures. Appl., 61 (1982), 41-63.  Google Scholar

[12]

Comm. P.D.E., 6 (1981), 883-901. doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

Commun. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[14]

Journal of Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[15]

Comm. P.D.E., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[16]

Proceedings of the Royal Society of Edinburgh, 138 (2008), 339-359. doi: 10.1017/S0308210506000394.  Google Scholar

[17]

Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[18]

Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[19]

Trans. Amer. Math. Soc., 346 (1994), 117-135. doi: 10.1090/S0002-9947-1994-1270664-3.  Google Scholar

[20]

Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[21]

SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301.  Google Scholar

[22]

Journal d'Analyse Mathématique, 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[23]

Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[24]

Differential Integral Equations, 12 (1999), 601-612.  Google Scholar

[25]

Comm. Pure and Appl, Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[26]

Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[28]

Diff. Int. Eq., 9 (1996), 465-479.  Google Scholar

[29]

Differential Integral Equations, 9 (1996), 1157-1164.  Google Scholar

[30]

Diff. Int. Eq., 9 (1996), 635-653.  Google Scholar

[31]

Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. Google Scholar

[32]

Comm. P.D.E., 23 (1998), 577-599. doi: 10.1080/03605309808821356.  Google Scholar

[33]

Advances in Mathematics, 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[34]

Diff. Int. Eq., 8 (1995), 1911-1922.  Google Scholar

[35]

Adv. Diff. Eq., 1 (1996), 241-264.  Google Scholar

[36]

Calc. Var., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[37]

X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., ().   Google Scholar

[38]

Journal of Differential Equations, 254 (2013), 2173-2182. doi: 10.1016/j.jde.2012.11.021.  Google Scholar

show all references

References:
[1]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[2]

Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[3]

Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[4]

AIMS Book Series, vol. 4, 2010.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[6]

Comm. Pure and Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[7]

Comm. P.D.E., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[8]

J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[9]

Rev. Mat. Iberoamericana, 20 (2004), 67-86.  Google Scholar

[10]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.  Google Scholar

[11]

J. Math. Pures. Appl., 61 (1982), 41-63.  Google Scholar

[12]

Comm. P.D.E., 6 (1981), 883-901. doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

Commun. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[14]

Journal of Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[15]

Comm. P.D.E., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[16]

Proceedings of the Royal Society of Edinburgh, 138 (2008), 339-359. doi: 10.1017/S0308210506000394.  Google Scholar

[17]

Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[18]

Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[19]

Trans. Amer. Math. Soc., 346 (1994), 117-135. doi: 10.1090/S0002-9947-1994-1270664-3.  Google Scholar

[20]

Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[21]

SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301.  Google Scholar

[22]

Journal d'Analyse Mathématique, 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[23]

Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[24]

Differential Integral Equations, 12 (1999), 601-612.  Google Scholar

[25]

Comm. Pure and Appl, Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[26]

Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[28]

Diff. Int. Eq., 9 (1996), 465-479.  Google Scholar

[29]

Differential Integral Equations, 9 (1996), 1157-1164.  Google Scholar

[30]

Diff. Int. Eq., 9 (1996), 635-653.  Google Scholar

[31]

Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. Google Scholar

[32]

Comm. P.D.E., 23 (1998), 577-599. doi: 10.1080/03605309808821356.  Google Scholar

[33]

Advances in Mathematics, 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[34]

Diff. Int. Eq., 8 (1995), 1911-1922.  Google Scholar

[35]

Adv. Diff. Eq., 1 (1996), 241-264.  Google Scholar

[36]

Calc. Var., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[37]

X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., ().   Google Scholar

[38]

Journal of Differential Equations, 254 (2013), 2173-2182. doi: 10.1016/j.jde.2012.11.021.  Google Scholar

[1]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016
[2]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[3]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069
[4]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037
[5]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[6]

Nadezhda Maltugueva, Nikolay Pogodaev. Modeling of crowds in regions with moving obstacles. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021066
[7]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017
[8]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
[9]

Muberra Allahverdi, Harun Aydilek, Asiye Aydilek, Ali Allahverdi. A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1973-1991. doi: 10.3934/jimo.2020054
[10]

Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024
[11]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[12]

Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341
[13]

Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005
[14]

Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037
[15]

John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021023
[16]

Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075
[17]

Andreas Neubauer. On Tikhonov-type regularization with approximated penalty terms. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021027
[18]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[19]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[20]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences