Spectral asymptotics for inverse nonlinear Sturm-Liouville problems

Abstract

We consider the nonlinear Sturm-Liouville problem −u′′(t) + f(u(t), u′(t)) = λu(t), u(t) > 0, t ∈ I := (−1/2, 1/2), u(±1/2) = 0, where f(x, y) = |x|p−1x − |y|m, p > 1, 1 ≤ m < 2 are constants and λ > 0 is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in R+ ×Lq(I) (1 ≤ q < ∞) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue λ = λq(α) as α := ‖uλ‖q → ∞, where uλ is a solution associated with given λ > π2.

Cite this paper

@inproceedings{Shibata2009SpectralAF, title={Spectral asymptotics for inverse nonlinear Sturm-Liouville problems}, author={Tetsutaro Shibata}, year={2009} }