Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-12T22:46:00.305Z Has data issue: false hasContentIssue false
  • English
  • Français

On the maximum of solutions for a semilinear elliptic problem

Published online by Cambridge University Press:  14 November 2011

Guido Sweers
Guido Sweers
Affiliation:
Faculty of Mathematics and Informatics, Delft University of Technology, P.O. Box 356, 2600 AJ Delft, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we study some properties of a semilinear elliptic eigenvalue problem with nondefinite right-hand side. In the first part we show that every solution will have its maximum in some specified interval J. If the domain is inside a cone in ℝN with N > 1, then J is strictly smaller than in the one-dimensional case. In the second part we show, for bounded domains, that if the maximum is inside some subinterval of J, then for any eigenvalue there will be at most one solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 3-4 , 1988 , pp. 357 - 370
Copyright
Copyright © Royal Society of Edinburgh 1988

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

1Clément, Ph. and Sweers, G.. Existence and multiplicity results for a semilinear elliptic eigenvalue problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 97–121.Google Scholar
2Clément, Ph. and Sweers, G.. Existence et multiplicité des solutions d'un problème aux valeurs propres elliptique semilinéaire. C.R. Acad. Sci., Paris Ser. I Math. 302 (1986), 681–683.Google Scholar
3Cosner, C. and Schmitt, K.. A priori bounds for positive solutions of a semilinear elliptic equation. Proc. Amer. Math. Soc. 95 (1985) 47–50.CrossRefGoogle Scholar
4Dancer, E. N. and Schmitt, K.. On positive solutions of semilinear elliptic equations. Proc. Amer. Math. Soc. 101 (1987), 445–452.CrossRefGoogle Scholar
5Fife, P.. Semilinear elliptic boundary value problems with small parameters. Arch. Rational. Mech. Anal. 52 (1973), 205–232.CrossRefGoogle Scholar
6Gidas, B., Ni, W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
7Hess, P.. On multiple solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differential Equations 6 (1981), 951–961.CrossRefGoogle Scholar
8Modica, L.. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math. 38 (1985), 679–684.CrossRefGoogle Scholar
9Poisson, S. D.. Second mémoire sur la distribution de la chaleur dans les corps solides. J. I'École Polytechnique 12 (1823), 249–403.Google Scholar
10Protter, M. and Weinberger, H.. Maximum principles in differential equations (Englewood Cliffs N.J.: Prentice Hall, 1967).Google Scholar
11Rabinowitz, P. H.. Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23 (1973), 172–185.CrossRefGoogle Scholar
12Serrin, J.. Nonlinear equations of second order. American Mathematical Society Symposium in Partial Differential Equations, Berkeley, August 1971.Google Scholar
13Sperb, R.. Maximum principles and their applications. Math, in Science and Engineering 157 (New York: Academic Press, 1981).Google Scholar