Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems
The author considers a class of nonlinear Sturm-Liouville problems \[ (0.1)\quad -u''=\lambda r(x)u-F(x,u)u,\quad a<x<+\infty,\quad u(a)\cos \theta -u'(a)\sin \theta =0,\quad u\in L^ 2. \] It is easy to see that (0.1) admits zero as a trivial solution \(\forall \lambda\) and that the linearization around zero has purely continuous spectrum \([0,+\infty)\). For a special type of nonlinearity \(F(x,u)=\psi (w(x)| u|^{\sigma})\) with \(\psi (t)\geq pt^ q\), \(\psi '>0\), \(w'(0)\geq 0\), \(w'/w\) nondecreasing, the author proves the uniqueness of solutions having a prescribed number of nodes. From this result it is possible to deduce that there are infinitely many continuous curves of solutions, bifurcating from the origin parametrized by \(\lambda\) and characterized by their nodal properties. Further, for a more general class of nonlinearities the author proves a bifurcation theorem analogous to the preceding one, where the continuous curves are replaced by unbounded connected components.
scientific article; zbMATH DE number 4171390
scientific article