Liouville type theorem for nonlinear elliptic equation with general nonlinearity

Xiaohui Yu

doi:10.3934/dcds.2014.34.4947

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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

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

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[18]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465. Google Scholar
[29]	B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157. Google Scholar
[30]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635. Google Scholar
[31]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[34]	S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911. Google Scholar
[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. Google Scholar
[36]	X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75. 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]	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. Google Scholar*

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series, (2010). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, Rev. Mat. Iberoamericana, 20 (2004), 67. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[18]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition,, Differential Integral Equations, 7 (1994), 301. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	Y. Lou and M. Zhu, Classifications of nonnegative solutions to some elliptic problems,, Differential Integral Equations, 12 (1999), 601. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar*
[28]	E. Mitidieri, Nonexistence of positive solutions of semilinear systems in $ R^N$,, Diff. Int. Eq., 9 (1996), 465. Google Scholar
[29]	B. Ou, Positive harmonic functions on the upper half space satisfying a nonlinear boundary condition,, Differential Integral Equations, 9 (1996), 1157. Google Scholar
[30]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, Diff. Int. Eq., 9 (1996), 635. Google Scholar
[31]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[34]	S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions,, Diff. Int. Eq., 8 (1995), 1911. Google Scholar
[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. Google Scholar
[36]	X. Yu, Liouville type theorems for integral equations and integral systems,, Calc. Var., 46 (2013), 75. 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]	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. Google Scholar*

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