Characterisation of bifurcation from the continuous spectrum by nodal properties (Q798820)

By variational methods it was known that the nonlinear eigenvalue problem \(-u''+w(x)| u|^{\sigma}u=\lambda u, u(0)=0\), \(u\in L^ 2(0,\infty)\) has infinitely many solutions for each \(\lambda >0\), if w is positive, \(\sigma >0\) and \(\int^{\infty}_{0}w^{- 2/\sigma}(x)dx<\infty\). Here it is shown for exponentially growing w that these solutions can be characterised by the number of zeros. For \(\lambda \to 0\) all solutions bifurcate from the lowest point of the continuous spectrum and this characterisation gives that there are infinitely many branches characterised by nodal properties. The proof is based on a dynamical systems approach which was recently extended to semilinear equations in \({\mathbb{R}}^ n\).