Liouville type theorem for nonlinear elliptic equation with general nonlinearity

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November 2014, 34(11): 4947-4966. doi: 10.3934/dcds.2014.34.4947

Liouville type theorem for nonlinear elliptic equation with general nonlinearity

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

Received September 2013 Revised December 2013 Published May 2014

Abstract
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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 - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947

References:

[1]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.
[2]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[3]	W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5.
[4]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010).
[5]	W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.
[7]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. P.D.E., 30 (2005), 59. doi: 10.1081/PDE-200044445.
[8]	M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb R_+^n$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429. doi: 10.1006/jmaa.1998.5958.
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67.
[10]	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.
[11]	D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation,, J. Math. Pures. Appl., 61 (1982), 41.
[12]	B. Gidas and J. Spruk, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. P.D.E., 6 (1981), 883. doi: 10.1002/cpa.3160340406.
[13]	B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Commun. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.
[14]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$,, Journal of Differential Equations, 225* (2006), 685. doi: 10.1016/j.jde.2005.10.016.*
[15]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^N$,, Comm. P.D.E., 33 (2008), 263. doi: 10.1080/03605300701257476.
[16]	Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $\mathbb R^N$ and in $\mathbb R^N_+$,, Proceedings of the Royal Society of Edinburgh, 138* (2008), 339. doi: 10.1017/S0308210506000394.*
[17]	F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1. doi: 10.1016/j.anihpc.2007.03.006.
[18]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.
[19]	B. Hu and H. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346* (1994), 117. doi: 10.1090/S0002-9947-1994-1270664-3.*
[20]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134* (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.*
[21]	C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.
[22]	Y. Li and L. Zhang, Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations,, Journal d'Analyse Mathématique, 90 (2003), 27. doi: 10.1007/BF02786551.
[23]	Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8.
[24]	Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601.
[25]	L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure and Appl, 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[26]	C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226* (2011), 2676. doi: 10.1016/j.aim.2010.07.020.*
[27]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195* (2010), 455. doi: 10.1007/s00205-008-0208-3.*
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465.
[29]	B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157.
[30]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635.
[31]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369.
[32]	J. Serrin and H. Zou, Existence of positive entire solutions of elliptic Hamiltonian systems,, Comm. P.D.E., 23 (1998), 577. doi: 10.1080/03605309808821356.
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Advances in Mathematics, 221* (2009), 1409. doi: 10.1016/j.aim.2009.02.014.*
[34]	S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911.
[35]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Diff. Eq., 1 (1996), 241.
[36]	X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75. doi: 10.1007/s00526-011-0474-z.
[37]	X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., ().
[38]	X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity,, Journal of Differential Equations, 254* (2013), 2173. doi: 10.1016/j.jde.2012.11.021.*

show all references

References:

[1]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.
[2]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Disc. Cont. Dyn. Sys., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[3]	W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, Acta Mathematica Scientia, 29 (2009), 949. doi: 10.1016/S0252-9602(09)60079-5.
[4]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010).
[5]	W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.
[7]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. P.D.E., 30 (2005), 59. doi: 10.1081/PDE-200044445.
[8]	M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb R_+^n$ with a nonlinear boundary condition,, J. Math. Anal. Appl., 223 (1998), 429. doi: 10.1006/jmaa.1998.5958.
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67.
[10]	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.
[11]	D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equation,, J. Math. Pures. Appl., 61 (1982), 41.
[12]	B. Gidas and J. Spruk, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. P.D.E., 6 (1981), 883. doi: 10.1002/cpa.3160340406.
[13]	B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via maximum principle,, Commun. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.
[14]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$,, Journal of Differential Equations, 225* (2006), 685. doi: 10.1016/j.jde.2005.10.016.*
[15]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^N$,, Comm. P.D.E., 33 (2008), 263. doi: 10.1080/03605300701257476.
[16]	Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in $\mathbb R^N$ and in $\mathbb R^N_+$,, Proceedings of the Royal Society of Edinburgh, 138* (2008), 339. doi: 10.1017/S0308210506000394.*
[17]	F. Hang, X. Wang and X. Yan, An integral equation in conformal geometry,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1. doi: 10.1016/j.anihpc.2007.03.006.
[18]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301.
[19]	B. Hu and H. Yin, The profile near blowup time for solution of the heat equation with a nonlinear boundary condition,, Trans. Amer. Math. Soc., 346* (1994), 117. doi: 10.1090/S0002-9947-1994-1270664-3.*
[20]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134* (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.*
[21]	C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049. doi: 10.1137/080712301.
[22]	Y. Li and L. Zhang, Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations,, Journal d'Analyse Mathématique, 90 (2003), 27. doi: 10.1007/BF02786551.
[23]	Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8.
[24]	Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601.
[25]	L. Ma and D. Chen, A Liouville type theorem for an integral system,, Comm. Pure and Appl, 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[26]	C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Advances in Mathematics, 226* (2011), 2676. doi: 10.1016/j.aim.2010.07.020.*
[27]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Ration. Mech. Anal., 195* (2010), 455. doi: 10.1007/s00205-008-0208-3.*
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465.
[29]	B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157.
[30]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635.
[31]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369.
[32]	J. Serrin and H. Zou, Existence of positive entire solutions of elliptic Hamiltonian systems,, Comm. P.D.E., 23 (1998), 577. doi: 10.1080/03605309808821356.
[33]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Advances in Mathematics, 221* (2009), 1409. doi: 10.1016/j.aim.2009.02.014.*
[34]	S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911.
[35]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Diff. Eq., 1 (1996), 241.
[36]	X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75. doi: 10.1007/s00526-011-0474-z.
[37]	X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., ().
[38]	X. Yu, Liouville type theorem in the Heisenberg group with general nonlinearity,, Journal of Differential Equations, 254* (2013), 2173. doi: 10.1016/j.jde.2012.11.021.*

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