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Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents

Published online by Cambridge University Press:  20 November 2018

Dongsheng Kang
Dongsheng Kang*
Affiliation:
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, P.R. Chinae-mail: dongshengkang@yahoo.com.cn
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Abstract

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In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy–Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Keywords

35B3335B4035J60elliptic problemsolutionHardy–Sobolev inequalitysingularityMoser iteration
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 537 - 547
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Abdellaoui, B., Felli, V., and Peral, I., Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole ℝ N. Adv. Differential Equations 9(2004), no. 5-6, 481–508.Google Scholar
[2] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Functional Analysis 14(1973), 349–381. http://dx.doi.org/10.1016/0022-1236(73)90051-7 Google Scholar
[3] Brézis, H. and Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36(1983) no. 4, 437–477. http://dx.doi.org/10.1002/cpa.3160360405 Google Scholar
[4] Caffarelli, L., Kohn, R., and Nirenberg, L., First order interpolation inequality with weights. Compositio Math. 53(1984), no. 3, 259–275.Google Scholar
[5] Cao, D. and Han, P., Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differential Equations 224(2006), no. 2, 332–372. http://dx.doi.org/10.1016/j.jde.2005.07.010 Google Scholar
[6] Catrina, F. and Wang, Z.-Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extermal functions. Comm. Pure Appl. Math. 54(2001), no. 2, 229–258. http://dx.doi.org/10.1002/1097-0312(200102)54:2h229::AID-CPA4i3.0.CO;2-I Google Scholar
[7] Chou, K. and Chu, C., On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc. 48(1993), no. 1, 137–151. http://dx.doi.org/10.1112/jlms/s2-48.1.137 Google Scholar
[8] Dautray, R. and Lions, P., Mathematical Analysis and Numerical Methods for Science and Technology. I. Physical Origins and Classical Methods. Springer-Verlag, Berlin, 1990.Google Scholar
[9] Felli, V. and Terracini, S., Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Comm. Partial Differential Equations 31(2006), no. 1-3, 469–495. http://dx.doi.org/10.1080/03605300500394439 Google Scholar
[10] Ghoussoub, N. and Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents,. Trans. Amer. Math. Soc. 352(2000), no. 12, 5703–5743. http://dx.doi.org/10.1090/S0002-9947-00-02560-5 Google Scholar
[11] Hardy, G., Littlewood, J., and Pólya, G., Inequalities. Cambridge University Press, Cambridge, UK, 1934.Google Scholar
[12] Huang, Y. and Kang, D., Elliptic systems involving the critical exponents and potentials. Nonlinear Anal. 71(2009), no. 9, 3638–3653. http://dx.doi.org/10.1016/j.na.2009.02.024 Google Scholar
[13] Jannelli, E., The role played by space dimension in elliptic critical problems. J. Differential Equations 156(1999), no. 2, 407–426. http://dx.doi.org/10.1006/jdeq.1998.3589 Google Scholar
[14] Kang, D. and Li, G., On the elliptic problems involving multi-singular inverse square potentials and multi-critical Sobolev-Hardy exponents. Nonlinear Anal. 66(2007) , no. 8, 1806–1816. http://dx.doi.org/10.1016/j.na.2006.02.026 Google Scholar
[15] Kang, D. and Peng, S., Positive solutions for singular critical elliptic problems. Appl. Math. Lett. 17(2004), no. 4, 411–416. http://dx.doi.org/10.1016/S0893-9659(04)90082-1 Google Scholar
[16] Lai, B. and Zhou, S., Asymptotic behavior of positive solutions of semilinear elliptic equations in RN. II. Acta Math. Sin. (Engl. Ser.) 26(2010), no. 9, 1723–1738. http://dx.doi.org/10.1007/s10114-010-8325-y Google Scholar
[17] Lions, P. L., The concentration compactness principle in the calculus of variations.T the limit case. I. Rev. Mat. Iberoamericana 1(1985). no. 1, 145–201.Google Scholar
[18] Lions, P. L., The concentration compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1(1985), no. 2, 45–121.Google Scholar
[19] Smets, D., Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities. Trans. Amer. Math. Soc. 357(2005), no. 7, 2909–2938. http://dx.doi.org/10.1090/S0002-9947-04-03769-9 Google Scholar
[20] Terracini, S., On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differential Equations 1(1996), no. 2, 241–264.Google Scholar