Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up


Authors: J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis
Journal: Proc. Amer. Math. Soc. 129 (2001), 3593-3602
MSC (2000): Primary 35J25; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9939-01-06229-3
Published electronically: June 6, 2001
MathSciNet review: 1860492
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we prove uniqueness of positive solutions to logistic singular problems $-\Delta u=\lambda(x) u -a(x) u^{p}$, $u_{\vert\partial \Omega }=+\infty $, $p>1$, $a>0$ in $\Omega$, where the main feature is the fact that $a_{\vert\partial\Omega}=0$. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near $\partial \Omega $. This expansion involves both the distance function $d(x)=\text{dist}(x,\partial \Omega)$ and the mean curvature $H$ of $\partial \Omega $.


References [Enhancements On Off] (What's this?)

  • 1. Bandle C., Essèn M., On the solutions of quasilinear elliptic problems with boundary blow-up, Sympos. Math. 35 (1994), 93-111. MR 95f:35077
  • 2. Bandle C., Marcus M., Sur les solutions maximales de problèmes elliptiques nonlinéaires: bornes isopérimetriques et comportement asymptotique, C. R. Acad. Sci. Paris Série I 311 (1990), 91-93. MR 91f:35096
  • 3. -, `Large' solutions of semilinear elliptic equations: Existence, uniqueness, and asymptotic behaviour, J. Anal. Math. 58 (1992), 9-24. MR 94c:35081
  • 4. -, On second order effects in the boundary behaviour of large solutions of semilinear elliptic problems, Differential Integral Equations 11 (1) (1998), 23-34. MR 98m:35049
  • 5. Bieberbach L., $\Delta u = e^{u}$ und die automorphen Funktionen, Math. Ann. 77 (1916), 173-212.
  • 6. Brézis H., Oswald L., Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55-64. MR 87c:35057
  • 7. Del Pino M., Letelier R., The influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal. (to appear) (1999).
  • 8. Díaz G., Letelier R., Unicidad de soluciones locales en algunas ecuaciones semilineales, Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics, Univ. Málaga, Málaga. (1989), 301-305. MR 91j:00015
  • 9. Díaz G., Letelier R., Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal. 20 (1993), 97-125. MR 94a:35017
  • 10. García-Melián J., Gómez-Reñasco R., López-Gómez J., Sabina de Lis J., Point-wise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Rational Mech. Anal. 145 (1998), 261-289. MR 2000b:35079
  • 11. D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd. edition, Springer Verlag, Berlin/New York, 1983. MR 86c:35035
  • 12. Keller J. B., On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math. 10 (1957), 503-510. MR 19:964c
  • 13. Kondrat'ev V. A., Nikishkin V. A., Asymptotics, near the boundary, of a solution of a singular boundary value problem for a semilinear elliptic equation, Differential Equations 26 (1990), 345-348. MR 91g:35048
  • 14. Lazer A. C., McKenna P. J., On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 (1993), 327-335. MR 95b:35070
  • 15. -, Asymptotic behaviour of solutions of boundary blow-up problems, Differential Integral Equations 7 (1994), 1001-1019. MR 95c:35084
  • 16. Loewner C., Nirenberg L., Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, p. 245-272. MR 50:10543
  • 17. Marcus M., Véron L., Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (2) (1997), 237-274. MR 97m:35068
  • 18. Osserman R., On the inequality $\Delta u \ge f(u)$, Pacific J. Math. 7 (1957), 1641-1647. MR 20:4701
  • 19. Ratto A., Rigoli M., Véron L., Scalar Curvature and Conformal Deformation of Hyperbolic Space, J. Funct. Anal. 121 (1994), 15-77. MR 95a:53062
  • 20. Véron L., Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math. 59 (1992), 231-250. MR 94k:35113

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J25, 35B40

Retrieve articles in all journals with MSC (2000): 35J25, 35B40


Additional Information

J. García-Melián
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, Astrofísico Francisco Sánchez s/n, 38271-La Laguna, Spain
Email: jjgarmel@ull.es

R. Letelier-Albornoz
Affiliation: Departamento de Matemáticas, Universidad de Concepción, Casilla 3-C, Concepción, Chile
Email: rletelie@gauss.cfm.udec.cl

J. Sabina de Lis
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, Astrofísico Francisco Sánchez s/n, 38271-La Laguna, Spain
Email: josabina@ull.es

DOI: https://doi.org/10.1090/S0002-9939-01-06229-3
Keywords: Boundary blow-up, uniqueness, sub and supersolutions, distance function
Received by editor(s): April 17, 2000
Published electronically: June 6, 2001
Additional Notes: This work was supported by DGES, project PB96-0621 (Spain) and grant FONDECYT No. 1000333 (Chile).
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society