Remarks on Brezis-Nirenberg theorem. (Q2712488)

Let \(E\) be a Banach space and \(f\in C^1(E, R)\) satisfy P-S condition. \(Q\) is a closed Banach manifold with boundary \(\partial Q\). \(\Gamma = \{p\in C(Q,E): p| _{\partial Q} = id\}\), \(c = \inf_{p\in\Gamma}\sup_{x\in Q} f(p(x))\). The main problem in critical points theory is to find critical points of \(f\). There are some results on this problem. In particular, \textit{H. Brèzis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 44, No. 8--9, 939--963 (1991; Zbl 0751.58006)] proved that \(c\) is a critical value of \(f\) if \(f\) meets the following condition:NEWLINENEWLINE\(Q\) is compact and for any \(p\in\Gamma\), there is a \(x\in Q\setminus\partial Q\) such that \(p(x)\notin\partial Q\), \(f(p(x))\geq c_1 = sup_{x\in\partial Q}f(x).\)NEWLINENEWLINEIn this paper, the author proves that the Brezis-Nirenberg condition is equivalent to some previous ones under some suitable assumptions on the \(K_c\).