Abstract
Let \(B_R \subset R^N (N\geq 3)\) be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem \(-\Delta u=\frac{\mu u}{|x|^2}+u^{2^*-1-\varepsilon}, u > 0 \) in \(B_R, u=0\) on ∂BR when ɛ→+ for suitable positive numbers μ
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Garcia Azorero, J.P., Peral, I.: Alonso, Hardy inequalities and some critical elliptic and parabolic problems. J. Diff. Eqns. 144, 441–476 (1998)
Atkinson, F.V., Brezis, H., Peletier, L.A.: Nodal solutions of elliptic equations with critical Sobolev exponent. J. Diff. Eqns. 85, 151–170 (1990)
Atkinson, F.V., Peletier, L.A.: Elliptic equations with nearly critical growth. J. Diff. Eqns. 70, 349–365 (1987)
Badiale, M., Tarantello, G.: A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)
Brezis, H., Peletier, L.A.: Asymptotics for elliptic equations involving critical Sobolev exponent. In: Colombini, F., Marino, A., Modica L., Spagnolo, S. (eds.) Partial Differential Equations and the Calculus of Variations, Basel, Birkhauser (1989)
Chen, X., Lu, G.: Asymptotics of radial osciliatory solutions of semilinear elliptic equations. Diff. Int. Eqns. 14, 1367–1380 (2001)
Cao, D., Peng, S.J.: A global compactness result for singular elliptic problems involving critical Sobolev exponent. Proc. Am. Math. Soc. 131, 1857–1866 (2003)
Catrina, F., Wang, Z.Q.: On the Caffarelli—Kohn—Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extermal functions. Comm. Pure Appl. Math. 53, 1–30 (2000)
Chou, K.S., Chu, C.W.: On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc. 48(2), 137–151 (1993)
Dupaigen, L.: A nonlinear elliptic PDE with the inverse potential. J. D'anal. Math. 86, 359–398 (2002)
Ekeland, I., Ghoussoub, N.: Selected new aspects of the calculus of variations in the large. Bull. Am. Math. Soc. 39, 207–265 (2002)
Ferrero, A., Gazzola, F.: Existence of solutions for singular critical growth semilinear elliptic equations. J. Diff. Eqns. 177, 494–522 (2001)
Hebey, E.: Asymptotics for some quasilinear elliptic equations. Diff. Inte. Eqns. 9, 71–88 (1996)
Hebey, E.: The asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth. Diff. Inte. Eqns. 13, 1073–1080 (2000)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via maximum principle. Comm. Math. Phys. 68, 209–243 (1979)
Jannelli, E.: The role played by space dimension in elliptic critcal problems. J. Diff. Eqns. 156, 407–426 (1999)
Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 352, 5703–5743 (2000)
Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a Chemotaxis system. J. Diff. Eqns. 72, 1–27 (1988)
Merle, F., Peletier, L.V.: Asymptotic behavior of positive solutions of elliptic equations with critical and supercritical growth I. the radial case. Arch. Ration. Mech. Anal. 112, 1–19 (1990)
Rey, O.: Proof of two conjectures of H. Brezis, L. A. Peletier. Manuscripta Math. 65, 19–37 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000) 35J60, 35B33
Rights and permissions
About this article
Cite this article
Cao, D., Peng, S. Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth. Annali di Matematica 185, 189–205 (2006). https://doi.org/10.1007/s10231-005-0150-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-005-0150-z