Boundary value problems for a semilinear elliptic equation with singular nonlinearity

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March 2016, 15(2): 399-412. doi: 10.3934/cpaa.2016.15.399

Boundary value problems for a semilinear elliptic equation with singular nonlinearity

Zongming Guo ^1, and Yunting Yu ^1,

1.	Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

Received March 2015 Revised October 2015 Published January 2016

Abstract
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Structure of solutions of boundary value problems for a semilinear elliptic equation with singular nonlinearity is studied. It is seen that the structure of solutions relies on the boundary values. The global branches of solutions of the boundary value problems are established. Moreover, some Liouville type results for the entire solutions of the equation are also obtained.

Keywords: boundary value problems, branches of solutions, singular nonlinearities, Liouville type results., Semilinear elliptic equations.

Mathematics Subject Classification: Primary: 35B45; Secondary: 35J4.

Citation: Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure and Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399

References:

[1]	P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., LX (2007), 1731-1768. doi: 10.1002/cpa.20189.
[2]	G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187.
[3]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math.Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803.
[4]	Z. M. Guo and L. Ma, Finite Morse index steady states of van der Waals force driven thin film equations, J. Math. Anal. Appl., 368 (2010), 559-572. doi: 10.1016/j.jmaa.2010.04.012.
[5]	Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity, J. Math. Anal.Appl., 354 (2009), 134-146. doi: 10.1016/j.jmaa.2009.01.001.
[6]	Z. M. Guo and J. C. Wei, Ausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.
[7]	Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. R. Soc. Edinburgh, 137A (2007), 963-994. doi: 10.1017/S0308210505001083.
[8]	Z. M. Guo and J. C. Wei, The Cauchy problem for a reaction-diffusion equation with a singular nonlinearity, J. Differential Equations, 240 (2007), 279-323. doi: 10.1016/j.jde.2007.06.012.
[9]	Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular nonlinearity, J. London Math. Soc., 78 (2008), 21-35. doi: 10.1112/jlms/jdm121.
[10]	Z. M. Guo and D. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations, Pacific J. Math., 236 (2008), 57-71. doi: 10.2140/pjm.2008.236.57.
[11]	H. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.
[12]	F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond., Ser. A Math. Phys. Eng. Sci., 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.
[13]	L. Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017.
[14]	J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.
[15]	J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC press, 2002.

show all references

References:

[1]	P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., LX (2007), 1731-1768. doi: 10.1002/cpa.20189.
[2]	G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187.
[3]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math.Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803.
[4]	Z. M. Guo and L. Ma, Finite Morse index steady states of van der Waals force driven thin film equations, J. Math. Anal. Appl., 368 (2010), 559-572. doi: 10.1016/j.jmaa.2010.04.012.
[5]	Z. M. Guo and X. Z. Peng, On the structure of positive solutions to an elliptic problem with a singular nonlinearity, J. Math. Anal.Appl., 354 (2009), 134-146. doi: 10.1016/j.jmaa.2009.01.001.
[6]	Z. M. Guo and J. C. Wei, Ausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120 (2006), 193-209. doi: 10.1007/s00229-006-0001-2.
[7]	Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. R. Soc. Edinburgh, 137A (2007), 963-994. doi: 10.1017/S0308210505001083.
[8]	Z. M. Guo and J. C. Wei, The Cauchy problem for a reaction-diffusion equation with a singular nonlinearity, J. Differential Equations, 240 (2007), 279-323. doi: 10.1016/j.jde.2007.06.012.
[9]	Z. M. Guo and J. C. Wei, Infinitely many turning points for an elliptic problem with a singular nonlinearity, J. London Math. Soc., 78 (2008), 21-35. doi: 10.1112/jlms/jdm121.
[10]	Z. M. Guo and D. Ye and F. Zhou, Existence of singular positive solutions for some semilinear elliptic equations, Pacific J. Math., 236 (2008), 57-71. doi: 10.2140/pjm.2008.236.57.
[11]	H. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.
[12]	F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond., Ser. A Math. Phys. Eng. Sci., 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816.
[13]	L. Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017.
[14]	J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.
[15]	J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC press, 2002.

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