Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems (Q803415)

The paper treats eigenvalues of semilinear elliptic problems as Lagrange multipliers from the variational statement: \(\gamma (t)=\sup_{\| u\|^ 2=t}g(u).\) Here g is an absolutely continuous Fréchet differentiable functional on a Hilbert space. The value function \(\gamma\) is increasing, continuous and has left and right hand derivatives \(\gamma '_{\pm}(t)\). Moreover, for every t there exist a solution (\(\rho\),u) to \(g'(u)=\rho u\) with \(\| u\|^ 2=t\) and \(\rho =2\gamma '_+(t)\) (or \(2\gamma '_ - (t))\), unless \(\gamma '_+(t)=0\) (or \(\gamma '_ -(t)=0).\) The paper estimates \(\gamma\) (t) with regard to such problems as \[ -\rho \Delta u=f(x,u),\quad u|_{\partial \Omega}=0,\quad \Omega \subset {\mathbb{R}}^ n. \]