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. 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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