Multiple Solutions and Bifurcation for a Class of Nonlinear Sturm-Liouville Eigenvalue Problems on an Unbounded Domain

摘要

A class of nonlinear Sturm-Liouville problems is considered. These problems admit zero as a trivial solution and the nonlinear operator linearized about zero has a purely continuous spectrum 0, INFINITY). Variational methods and approximation arguments are used to obtain the existence of nontrivial solutions with any prescribed number of nodes and for some nonlinearities it is shown that this solution is unique. Moreover, the lowest point of the continuous spectrum is bifurcation point; infinitely many continua of solutions, which are distinguished by nodal properties, bifurcate from the line of trivial solutions at this point. Results are also obtained in higher dimensions via investigation of the set of radial solutions of appropriate partial differential equations. Keywords: Nodes; Ordinary differential equations; Boundary value problems. (KR)