# American Institute of Mathematical Sciences

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. [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. [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. [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. 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. doi: 10.1142/S0219199702000737. [7] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Maximum Principle and Applications, (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. 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. doi: 10.1017/S0024610701002289. [10] S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Functional Analysis, 192 (2002), 186. 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. 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. 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. 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. [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. [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. [17] C. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum, (1992). [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.

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. [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. [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. [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. 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. doi: 10.1142/S0219199702000737. [7] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Maximum Principle and Applications, (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. 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. doi: 10.1017/S0024610701002289. [10] S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Functional Analysis, 192 (2002), 186. 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. 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. 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. 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. [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. [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. [17] C. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum, (1992). [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.
 [1] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111 [2] Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033 [3] Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1 [4] Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017 [5] Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363 [6] Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031 [7] Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513 [8] Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212 [9] Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377 [10] Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675 [11] Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure & Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175 [12] Huan Gao, Zhibao Li, Haibin Zhang. A fast continuous method for the extreme eigenvalue problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1587-1599. doi: 10.3934/jimo.2017008 [13] Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31 [14] Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367 [15] Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 [16] John Villavert. Sharp existence criteria for positive solutions of Hardy--Sobolev type systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 493-515. doi: 10.3934/cpaa.2015.14.493 [17] Li-Fang Dai, Mao-Lin Liang, Wei-Yuan Ma. Optimization problems on the rank of the solution to left and right inverse eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (1) : 171-183. doi: 10.3934/jimo.2015.11.171 [18] Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057 [19] Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747 [20] Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-15. doi: 10.3934/jimo.2018049

2017 Impact Factor: 1.179