## Differential and Integral Equations

### $L^q$ spectral asymptotics for nonlinear Sturm-Liouville problems

Tetsutaro Shibata

#### Abstract

We consider the nonlinear Sturm-Liouville problem $$-u''(t) + f(u(t)) = \lambda u(t), \ \ u(t) > 0, \quad t \in I := (0, 1), \ \ u(0) = u(1) = 0,$$ where $\lambda > 0$ is an eigenvalue parameter. For better understanding of the global behavior of the branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q \le \infty$), we establish precise asymptotic formulas for the eigenvalue $\lambda$ with respect to $\Vert u_\lambda\Vert_q$, where $u_\lambda$ is the unique solution associated with given $\lambda > \pi^2$.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 7 (2006), 773-783.

Dates
First available in Project Euclid: 21 December 2012

Shibata, Tetsutaro. $L^q$ spectral asymptotics for nonlinear Sturm-Liouville problems. Differential Integral Equations 19 (2006), no. 7, 773--783. https://projecteuclid.org/euclid.die/1356050349