Spectral asymptotics for inverse nonlinear Sturm-Liouville problems

T. Shibata, Hiroshima University, Higashi-Hiroshima, Japan

E. J. Qualitative Theory of Diff. Equ., No. 58. (2009), pp. 1-18.

Abstract: We consider the nonlinear Sturm-Liouville problem
$$
-u''(t) + f(u(t), u'(t)) = \lambda u(t),
\enskip u(t) > 0,
\quad t \in I := (-1/2, 1/2), \enskip u(\pm 1/2) = 0,
$$
where $f(x, y) = \vert x\vert^{p-1}x - \vert y\vert^m$, $p > 1, 1 \le m < 2$ are constants and $\lambda > 0$ is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q < \infty$) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue $\lambda = \lambda_q(\alpha)$ as $\alpha :=\Vert u_\lambda\Vert_q \to \infty$, where $u_\lambda$ is a solution associated with given $\lambda > \pi^2$.

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