Local behavior of solutions of some elliptic equations. (English) Zbl 0617.35040
We study the local behavior of solutions of some nonlinear elliptic equations. These equations are of interest in differential geometry and mathematical physics. Our main result reads as follows.
Theorem. Let \(u\in C^ 2\) (B\(| \{0\})\) be a nonnegative solution of \[ \Delta u+| x|^{\sigma} u^{(n+\sigma)/(n-2)}=0\quad in\quad B| \{0\}, \] where \(B=\{x\in {\mathbb{R}}^ n:| x| <1\), \(n\geq 3\}\) and \(-2<\sigma <2\). Then u has either a removable singularity at \(\{\) \(0\}\) or \[ \lim_{| x| \to 0} | x|^{(n-2)}(-\ln | x|)^{(n-2)/(\sigma +2)}u(x)=\ell \] exists and \(\ell =((n- 2)/(\sigma +2))1/(\sigma +2))^{(n-2)}\). Furthermore solutions with the behavior above exist.
Theorem. Let \(u\in C^ 2\) (B\(| \{0\})\) be a nonnegative solution of \[ \Delta u+| x|^{\sigma} u^{(n+\sigma)/(n-2)}=0\quad in\quad B| \{0\}, \] where \(B=\{x\in {\mathbb{R}}^ n:| x| <1\), \(n\geq 3\}\) and \(-2<\sigma <2\). Then u has either a removable singularity at \(\{\) \(0\}\) or \[ \lim_{| x| \to 0} | x|^{(n-2)}(-\ln | x|)^{(n-2)/(\sigma +2)}u(x)=\ell \] exists and \(\ell =((n- 2)/(\sigma +2))1/(\sigma +2))^{(n-2)}\). Furthermore solutions with the behavior above exist.
Keywords:
Yang-Mills-Higgs theory; astrophysics; asymptotic behavior; nonlinear elliptic equations; removable singularityReferences:
| [1] | Aviles, P.: On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J.35, 773-791 (1983) · Zbl 0548.35042 · doi:10.1512/iumj.1983.32.32051 |
| [2] | Chandrasekar, S.: An introduction to the study of stellar structure. Chicago, IL: The University of Chicago Press, 1939 |
| [3] | Fowler, R. H.: Further studies of Emden’s and similar differential equations, Q. J. Math. Oxf. II Ser.2, 259-287 (1931) · Zbl 0003.23502 · doi:10.1093/qmath/os-2.1.259 |
| [4] | Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math.4, 525-598 (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406 |
| [5] | Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear equations. Commun. Partial Differ. Equations, 883-901 (1981) · Zbl 0462.35041 |
| [6] | Giga, Y., Kohn, R.: Asymptotically self-similar blow up of semilinear heat equations Commun. Pure Appl. Math. (1985) · Zbl 0585.35051 |
| [7] | Hopf, E.: On Edmen’s differential equation, Roy. Astron. Soc. Monthly Notice91, 653-663 (1931) · Zbl 0002.10301 |
| [8] | Ni, W-M, Serrin, J.: in preparation |
| [9] | Sibner, L., Sibner, R.: Removable singularities of coupled Yang-Mills fields in ?3, Commun. Math. Phys.93, 1-17 (1984) · Zbl 0552.35028 · doi:10.1007/BF01218636 |
| [10] | Simon, L.: Asymptotics for a class of nonlinear evolution equations with applications to geometric problems. Ann. Math.118, 525-571 (1983) · Zbl 0549.35071 · doi:10.2307/2006981 |
| [11] | Veron, L.: Comportement asymptotique des solutions d’équations elliptiques semilinéaries dans ? n . Ann. Mat. Pura Appl.CXXVII, 25-50 (1981) · Zbl 0467.35013 · doi:10.1007/BF01811717 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
