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)

 
 

 

A priori bounds for positive solutions of a semilinear elliptic equation


Authors: Chris Cosner and Klaus Schmitt
Journal: Proc. Amer. Math. Soc. 95 (1985), 47-50
MSC: Primary 35J60; Secondary 35B45
DOI: https://doi.org/10.1090/S0002-9939-1985-0796444-0
MathSciNet review: 796444
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the semilinear elliptic equation $ - \Delta u = f(u)$, $ x \in \Omega $, subject to zero Dirichlet boundary conditions, where $ \Omega \subset {{\mathbf{R}}^n}$ is a bounded domain with smooth boundary and the nonlinearity $ f$ assumes both positive and negative values. Under the assumption that $ \Omega $ satisfies certain symmetry conditions we establish two results providing lower bounds on the $ {C^0}(\overline \Omega )$ norm of positive solutions. The bounds derived are the same one obtains in dimension $ n = 1$.


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

  • [1] A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411-422. MR 563992 (81j:35092)
  • [2] C. Bandle, Isoperimetric inequalities and applications, Pitman, Boston, Mass., 1980. MR 572958 (81e:35095)
  • [3] K. J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems, SIAM J. Math. Anal. 10 (1979), 875-883. MR 541087 (82k:35043)
  • [4] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 544879 (80h:35043)
  • [5] P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations 6 (1981), 951-961. MR 619753 (82j:35062)
  • [6] B. Kawohl, A geometric property of level sets of solutions to semilinear elliptic Dirichlet problems, Appl. Analysis 16 (1983), 229-233. MR 712734 (85f:35083)
  • [7] H. O. Peitgen, D. Saupe and K. Schmitt, Nonlinear elliptic boundary value problem versus their finite difference approximations: Numerically irrelevant solutions, J. Reine Angew. Math. 322 (1981), 74-117. MR 603027 (82h:65076)
  • [8] H. O. Peitgen and K. Schmitt, Global topological perturbations of nonlinear elliptic eigenvalue problems, Math. Appl. Sci. 5 (1983), 376-388. MR 716662 (84k:35069)
  • [9] R. Sperb, Maximum principles and their applications, Academic Press, New York, 1981. MR 615561 (84a:35033)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J60, 35B45

Retrieve articles in all journals with MSC: 35J60, 35B45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0796444-0
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society