Article Contents
Article Contents

# Isolated singularity for semilinear elliptic equations

• 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.
Mathematics Subject Classification: 35J61, 35B40.

 Citation:

•  [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, accepted. [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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