Oscillating global continua of positive solutions of semilinear elliptic problems
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- by Bryan P. Rynne PDF
- Proc. Amer. Math. Soc. 128 (2000), 229-236 Request permission
Abstract:
Let $\Omega$ be a bounded domain in $\mathbb {R}^n$, $n \ge 1$, with $C^2$ boundary $\partial \Omega$, and consider the semilinear elliptic boundary value problem \begin{align*} L u &= \lambda a u + g(\cdot ,u)u, \quad \text {in} \Omega ,\ u &= 0, \quad \text {on} \partial \Omega , \end{align*} where $L$ is a uniformly elliptic operator on $\bar {\Omega }$, $a \in C^0(\bar {\Omega })$, $a$ is strictly positive in $\bar {\Omega }$, and the function $g:\bar {\Omega } \times \mathbb {R} \to \mathbb {R}$ is continuously differentiable, with $g(x,0) = 0$, $x \in \bar {\Omega }$. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue $\lambda _1$ of the linear problem. We show that under certain oscillation conditions on the nonlinearity $g$, this continuum oscillates about $\lambda _1$, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each $\lambda$ in an open interval containing $\lambda _1$.References
- Rehana Bari and Bryan P. Rynne, The structure of Rabinowitz’ global bifurcating continua for problems with weak nonlinearities, Mathematika 44 (1997), no. 2, 419–433. MR 1600502, DOI 10.1112/S0025579300012717
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Marco Holzmann and Hansjörg Kielhöfer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Math. Ann. 300 (1994), no. 2, 221–241. MR 1299061, DOI 10.1007/BF01450485
- Hansjörg Kielhöfer and Stanislaus Maier, Infinitely many positive solutions of semilinear elliptic problems via sub- and supersolutions, Comm. Partial Differential Equations 18 (1993), no. 7-8, 1219–1229. MR 1233192, DOI 10.1080/03605309308820971
- S. Maier-Paape and K. Schmitt, Asymptotic behaviour of solution continua for semilinear elliptic problems, Can. Appl. Math. Quart. 4, (1996), 211–228.
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
- Bryan P. Rynne, Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter, Z. Angew. Math. Phys. 47 (1996), no. 5, 730–739 (English, with English and German summaries). MR 1420852, DOI 10.1007/BF00915272
- Bryan P. Rynne, The structure of Rabinowitz’ global bifurcating continua for generic quasilinear elliptic equations, Nonlinear Anal. 32 (1998), no. 2, 167–181. MR 1491622, DOI 10.1016/S0362-546X(97)00471-9
- Renate Schaaf and Klaus Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc. 306 (1988), no. 2, 853–859. MR 933322, DOI 10.1090/S0002-9947-1988-0933322-5
- R. Schaaf and K. Schmitt, Asymptotic behaviour of positive solution branches of elliptic problems with linear part at resonance, ZAMP 43, (1992), 645–676.
Additional Information
- Bryan P. Rynne
- Affiliation: Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland
- Email: bryan@ma.hw.ac.uk
- Received by editor(s): March 26, 1998
- Published electronically: May 27, 1999
- Communicated by: Lesley M. Sibner
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 229-236
- MSC (1991): Primary 35B32; Secondary 35B65
- DOI: https://doi.org/10.1090/S0002-9939-99-05168-0
- MathSciNet review: 1641097