A Necessary and Sufficient Condition for Palais--Smale Conditions

Abstract

In this paper we prove the following assertions: $(1)$ $\alpha _{I}=\alpha _{M}=\alpha _{\Gamma }=\alpha _{\Gamma ^{\prime }};$ $(2)$ let $\Omega _{0}=\Omega _{1}\cup \Omega _{2},$ where $\Omega _{1}\cap \Omega _{2}$ is bounded, and let $\alpha _{i}=\alpha (\Omega _{i})$ be the index of J in $\Omega _{i}$ for $i=0,1,2.$ J satisfies the $(PS)_{\alpha _{0}}$-condition if and only if the inequality $\alpha _{0} < \min \{\alpha _{1},\alpha _{2}\}$ holds; $(3)$ the union of a solvable domain and an unsolvable domain may be solvable and the union of two unsolvable domains may be solvable.