Bifurcation from the Continuous Spectrum in L^p(ℝ)

Abstract

The study of small solutions of the nonlinear eigenvalue problem:

$$ \Delta {\text{u(x)}}\, + \,\lambda {\text{u(x)}}\, + \,{\text{q(x)}}\,\left| {\text{u(x)}} \right|^\sigma {\text{u(x)}}\, = \,0 {\text { for x}}\, \in \,{\text{IR}}^{\text{N}} $$

$$ \eqalign{ & \lim \,{\text{u(x)}}\, = \,0 \cr & \left| {\text{x}} \right| \to \infty \cr} $$

where σ > 0 and q: ℝ^N → ℝ are given, leads to questions involving bifurcation from the continuous spectrum since its linearisation about u ≡ 0 has the interval [0,∞) as spectrum.