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

Isolated singularity for semilinear elliptic equations

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.
Citation: Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 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.  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.  doi: 10.1016/j.nonrwa.2007.02.010.  Google Scholar

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

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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.  doi: 10.1016/j.jfa.2010.03.015.  Google Scholar

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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.  doi: 10.1142/S0219199702000737.  Google Scholar

[7]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Maximum Principle and Applications, (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.  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.  doi: 10.1017/S0024610701002289.  Google Scholar

[10]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Functional Analysis, 192 (2002), 186.  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.  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.  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.  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.   Google Scholar

[15]

J. López-Gómez, The boundary blow-up rate of large solutions,, J. Differential Equations, 195 (2003), 25.  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.  doi: 10.1016/j.jde.2005.08.008.  Google Scholar

[17]

C. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum, (1992).   Google Scholar

[18]

D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials,, J. Differential Equations, 190 (2003), 524.  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.  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.  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, 347 (2009), 153.  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.  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.  doi: 10.1142/S0219199702000737.  Google Scholar

[7]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Maximum Principle and Applications, (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.  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.  doi: 10.1017/S0024610701002289.  Google Scholar

[10]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Functional Analysis, 192 (2002), 186.  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.  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.  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.  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.   Google Scholar

[15]

J. López-Gómez, The boundary blow-up rate of large solutions,, J. Differential Equations, 195 (2003), 25.  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.  doi: 10.1016/j.jde.2005.08.008.  Google Scholar

[17]

C. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum, (1992).   Google Scholar

[18]

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

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