Electron. J. Diff. Equ., Vol. 2009(2009), No. 107, pp. 1-11.

Inverse eigenvalue problems for semilinear elliptic equations

Tetsutaro Shibata

Abstract:
We consider the inverse nonlinear eigenvalue problem for the equation
$$\displaylines{ -\Delta u + f(u) = \lambda u, \quad u > 0 \quad \hbox{in } \Omega,\cr u = 0 \quad \hbox{on } \partial\Omega, }$$
where $f(u)$ is an unknown nonlinear term, $\Omega \subset \mathbb{R}^N$ is a bounded domain with an appropriate smooth boundary $\partial\Omega$ and $\lambda > 0$ is a parameter. Under basic conditions on $f$, for any given $\alpha > 0$, there exists a unique solution $(\lambda, u) = (\lambda(\alpha), u_\alpha) \in \mathbb{R}_+ \times C^2(\bar{\Omega})$ with $\|u_\alpha\|_2 = \alpha$. The curve $\lambda(\alpha)$ is called the $L^2$-bifurcation branch. Using a variational approach, we show that the nonlinear term $f(u)$ is determined uniquely by $\lambda(\alpha)$.

Submitted November 14, 2008. Published September 10, 2009.
Math Subject Classifications: 35P30.
Key Words: Inverse eigenvalue problems; nonlinear elliptic equation; variational method.

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Tetsutaro Shibata
Department of Applied Mathematics
Graduate School of Engineering
Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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