Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T02:25:35.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions

Published online by Cambridge University Press:  14 November 2011

C. Budd ,
M. C. Knaap  and
L. A. Peletier
C. Budd
Affiliation:
School of Mathematics, Bristol University, Bristol BS8ITW, U.K.
M. C. Knaap
Affiliation:
Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands
L. A. Peletier
Affiliation:
Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Synopsis

Asymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu = λ(up − uq) in the unit ball B in ℝN (N > 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Here p is the critical Sobolev exponent (N + 2)/(N − 2) and 0 < q < p − 1 = 4/(N − 2).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 225 - 250
Copyright
Copyright © Royal Society of Edinburgh 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V. and Peletier, L. A.. Emder–Fowler equations involving critical exponents. Nonlinear Anal. 10 (1986), 755–776.Google Scholar
2Atkinson, F. V. and Peletier, L. A.. Large solutions of elliptic equations involving critical exponents. Asymptotic Anal. 1 (1988), 139–160.CrossRefGoogle Scholar
3Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Fund. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
4, Adimurthi and Yadava, S. L.. Existence and nonexistence of positive radial solutions for Sobolev critical exponent problem with Neumann boundary condition (preprint).Google Scholar
5Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
6Budd, C.. Semilinear elliptic equations with near critical growth rates. Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 249–270.CrossRefGoogle Scholar
7Keller, E. and Segel, L.. Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970), 399–415.CrossRefGoogle ScholarPubMed
8Lin, C.-S. and Ni, W.-M.. On the diffusion coefficient of a semilinear Neumann problem. Trento Conference, Lecture Notes (Berlin: Springer, 1986).Google Scholar
9Lin, C.-S., Ni, W.-M. and Tagaki, I.. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations. 72 (1988), 1–27.CrossRefGoogle Scholar
10Ni, W.-M.. On the positive radial solutions of some semi-linear elliptic equations on ℝn. Appl. Math. Optim. 9 (1983), 373–380.CrossRefGoogle Scholar
11Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973), 161–201.CrossRefGoogle Scholar
12Schaaf, R.. Stationary solutions of chemotaxis systems. Trans. Amer. Math. Soc. 292 (1985), 531–556.Google Scholar