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

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].