On the perturbation of critical values of maximum-minimum type (Q686863)

The author studies \(C^ 1\)-functionals \(\Phi_*=\Phi+\Psi:M \to \mathbb{R}\) where \(M\) is either a Banach space \(X\) or a bounded hypersurface in \(X\). It is assumed that \(\Phi_*\) satisfies the Palais-Smale condition. If \(c\) is a minimax critical value of \(\Phi\) then there exists a corresponding minimax critical value \(c_*\) of \(\Phi_*\) which is close to \(c\) provided \(\Psi\) is small. The applications to Dirichlet problems of the form \(-\Delta u+p(x,u) =0\) are as in the book of \textit{P. H. Rabinowitz}, `Minimax Methods in Critical Point Theory with Applications to Differential Equations' (1986; Zbl 0609.58002). The minimax values defined in Theorem 3 are all equal to \(\sup \varphi\). Similarly the critical value defined in Theorem 4 is a local minimum.