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A Palais-Smale approach to Sobolev subcritical operators. (English) Zbl 1030.35032
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 H. C. Wang [Trans. Am. Math. Soc. 352, 4237-4256 (2000; Zbl 0951.35043)].

MSC:
 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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