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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

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 & 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.  Google Scholar

[2]

G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC press, 2002.  Google Scholar

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.  Google Scholar

[2]

G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and touchdown in electrostatic MEMS, Proceedings of ICMEMS, (2003), 182-187. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC press, 2002.  Google Scholar

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