## Differential and Integral Equations

### On Rabinowitz alternative for the Laplace-Beltrami operator on $S^{n-1}$: continua that meet infinity

Sławomir Rybicki

#### Abstract

Let $\Lambda$ be the Laplace--Beltrami operator on $S^{n-1}$. The aim of this paper is to prove that any continuum of nontrivial solutions of the equation $-\Lambda u = f(u,\lambda),$ which bifurcate from the set of trivial solutions, is unbounded in $H^1(S^{n-1}) \times R$. As the main tool we use degree theory for $S^1$--equivariant, gradient operators defined in [15].

#### Article information

Source
Differential Integral Equations, Volume 9, Number 6 (1996), 1267-1277.

Dates
First available in Project Euclid: 6 May 2013

https://projecteuclid.org/euclid.die/1367846900

Mathematical Reviews number (MathSciNet)
MR1409927

Zentralblatt MATH identifier
0879.35020

Subjects
Primary: 58E09: Group-invariant bifurcation theory
Secondary: 58G03

#### Citation

Rybicki, Sławomir. On Rabinowitz alternative for the Laplace-Beltrami operator on $S^{n-1}$: continua that meet infinity. Differential Integral Equations 9 (1996), no. 6, 1267--1277. https://projecteuclid.org/euclid.die/1367846900