%0 Journal Article
%T Inverse eigenvalue problems for semilinear elliptic equations
%A Tetsutaro Shibata
%J Electronic Journal of Differential Equations
%D 2009
%I Texas State University
%X 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 $lambda(alpha)$.
%K Inverse eigenvalue problems
%K nonlinear elliptic equation
%K variational method
%U http://ejde.math.txstate.edu/Volumes/2009/107/abstr.html