Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 107, pp. 1-11.
Title: Inverse eigenvalue problems for
semilinear elliptic equations
Authors: Tetsutaro Shibata (Hiroshima Univ., Japan)
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.