A Palais-Smale approach to Sobolev subcritical operators (Q1411254)

The domain \(\Omega\subset\mathbb R^N\) is an achieved one if there is \(u_0\in H^1_0(\Omega)\) such that \[ \|u_0\|_{L^p(\Omega)}/\|u_0\|_{H^1(\Omega)} = \sup \{\|u\|_{L^p(\Omega)}/\|u\|_{H^1(\Omega)}\mid u\in H^1_0(\Omega)\setminus\{0\}\}. \] The authors describe the achieved and some nonachieved domains for subcritical \(p\). See also the previous paper by \textit{H. C. Wang} [Trans. Am. Math. Soc. 352, 4237-4256 (2000; Zbl 0951.35043)].