Isolated singularity for semilinear elliptic equations

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239

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.
[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.
[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.
[4]	F. Cîrstea, A lcation of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, ().
[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.
[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.
[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.
[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.
[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.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Functional Analysis, 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.
[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.

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.
[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.
[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.
[4]	F. Cîrstea, A lcation of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, ().
[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.
[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.
[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.
[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.
[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.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Functional Analysis, 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	C. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.
[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.

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