American Institute of Mathematical Sciences

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July  2015, 35(7): 3239-3252. doi: 10.3934/dcds.2015.35.3239

Isolated singularity for semilinear elliptic equations

Lei Wei 1, and  Zhaosheng Feng 2,

1. 

School of Mathematical and Statistics, Jiangsu Normal University, Xuzhou 221116, China
2. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  December 2013 Revised  August 2014 Published  January 2015

In this paper, we study a class of semilinear elliptic equations with the Hardy potential. By means of the super-subsolution method and the comparison principle, we explore the existence of a minimal positive solution and a maximal positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin. Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.
Keywords: eigenvalue, minimal positive solution., Hardy potential, super-subsolution method.
Mathematics Subject Classification: 35J61, 35B4.
Citation: Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239
References:
[1]

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J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

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J. García-Melián, Boundary behavior for large solutions to elliptic equations with singular weights, Nonlinear Anal., 67 (2007), 818-826. doi: 10.1016/j.na.2006.06.041.  Google Scholar

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J. López-Gómez, The Maximum Principle and the Existence of Principal Eigenvalues for Some Linear Weighted Boundary Value Problems, J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.  Google Scholar

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J. López-Gómez, Large solutions, metasolutions and asymptotic behavior of a class of sublinear parabolic problems with refuges, Electronic J. Differential Equations, 5 (2000), 135-171. Google Scholar

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J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations, 195 (2003), 25-45. doi: 10.1016/j.jde.2003.06.003.  Google Scholar

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J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations, 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.  Google Scholar

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C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.  Google Scholar

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D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. doi: 10.1016/S0022-0396(02)00178-X.  Google Scholar

show all references

References:
[1]

B. Abdellaoui, I. Peral and A. Primo, Elliptic problems with a Hardy potential and critical growth in the gradient Non-resonance and blow-up results, J. Differential Equations, 239 (2007), 386-416. doi: 10.1016/j.jde.2007.05.010.  Google Scholar

[2]

P. Álvarez-Caudevilla and J. López-Gómez, Metasolutions in cooperative systems, Nonlinear Anal. Real World Appl., 9 (2008), 1119-1157. doi: 10.1016/j.nonrwa.2007.02.010.  Google Scholar

[3]

N. Chaudhuri and F. Cîrstea, On trichotomy of positive singular solutions associated with the Hardy-Sobolev operator, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 153-158. doi: 10.1016/j.crma.2008.12.018.  Google Scholar

[4]

F. Cîrstea, A lcation of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, ().   Google Scholar

[5]

F. Cîrstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations, J. Functional Analysis, 259 (2010), 174-202. doi: 10.1016/j.jfa.2010.03.015.  Google Scholar

[6]

F. Cîrstea and V. D. Rădulescu, Existence and uniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp. Math., 4 (2002), 559-586. doi: 10.1142/S0219199702000737.  Google Scholar

[7]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Maximum Principle and Applications, Vol. I, World Scientific Publishing, 2006. doi: 10.1142/9789812774446.  Google Scholar

[8]

Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31 (1999), 1-18. doi: 10.1137/S0036141099352844.  Google Scholar

[9]

Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar

[10]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Functional Analysis, 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.  Google Scholar

[11]

J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.  Google Scholar

[12]

J. García-Melián, Boundary behavior for large solutions to elliptic equations with singular weights, Nonlinear Anal., 67 (2007), 818-826. doi: 10.1016/j.na.2006.06.041.  Google Scholar

[13]

J. López-Gómez, The Maximum Principle and the Existence of Principal Eigenvalues for Some Linear Weighted Boundary Value Problems, J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.  Google Scholar

[14]

J. López-Gómez, Large solutions, metasolutions and asymptotic behavior of a class of sublinear parabolic problems with refuges, Electronic J. Differential Equations, 5 (2000), 135-171. Google Scholar

[15]

J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations, 195 (2003), 25-45. doi: 10.1016/j.jde.2003.06.003.  Google Scholar

[16]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations, 224 (2006), 385-439. doi: 10.1016/j.jde.2005.08.008.  Google Scholar

[17]

C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.  Google Scholar

[18]

D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. doi: 10.1016/S0022-0396(02)00178-X.  Google Scholar

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