Inverse eigenvalue problems for semilinear elliptic equations

Shibata, Tetsutaro

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

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 $\lambda(\alpha)$.

Détail

Classification : 35P30
Keywords: inverse eigenvalue problems, nonlinear elliptic equation, variational method