Inverse eigenvalue problems for semilinear elliptic equations

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 } partialOmega, } where $f(u)$ is an unknown nonlinear term, $Omega subset mathbb{R}^N$ is a bounded domain with an appropriate smooth boundary $partialOmega$ and $lambda > 0$ is a parameter. Under basic conditions on $f$, for any given $alpha > 0$, there exists a unique solution $(lambda, u) = (lambda(alpha), u_alpha) in mathbb{R}_+ imes C^2(ar{Omega})$ 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 $lambda(alpha)$.