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Nonlinear elliptic equations of critical Sobolev growth from a dynamical viewpoint. (English) Zbl 1210.35092

Bahri, Abbas (ed.) et al., Noncompact problems at the intersection of geometry, analysis, and topology. Proceedings of the conference on noncompact variational problems and general relativity held in honor of Haim Brezis and Felix Browder at Rutgers University, New Brunswick, NJ, USA, October 14–18, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3635-8/pbk). Contemporary Mathematics 350, 115-125 (2004).
Let \((M,g)\) be a smooth compact Riemannian manifold of dimension \(n \geq 3\). For \(\alpha > 0\) let \(S_{\alpha} \subset H_2^1\) be the set of solutions of \[ \Delta_gu + \alpha u = u^{2^{\ast}-1}, \] where \(2^{\ast} = \frac{2n}{n-2}\) is the critical Sobolev exponent. Let \[ E_m(\alpha) = \inf\{\| u\| _{L^{2^{\ast}}};\; u \in S_{\alpha}\}, \]
\[ E_{\Lambda} = \{u \in H_2^1;\; \| u\| _{L^{2^{\ast}}} \leq \Lambda\}, \] and \[ S_{\Lambda} = \{\alpha;\; S_{\alpha}\cap E_{\Lambda} \neq \emptyset\} \] for \(\Lambda > 0\). The following questions are addressed in the paper:
- Given \(\Lambda > 0\), is \(S_{\Lambda}\) bounded, closed in \((0,\infty)\), and/or connected?
- Is \(E_m\) a continuous and/or nondecreasing function of \(\alpha\)? Is it regular, convex or concave?
- Does \(E_m(\alpha) \to \infty\) as \(\alpha \to \infty\)? What is the asymptotic behaviour of \(E_m\) as \(\alpha \to \infty\)?
- Given \(\Lambda > 0\), is \(\big(\bigcup\limits_{\alpha\geq 0}S_{\alpha}\bigr)\cap E_{\Lambda}\) compact in the \(C^2\)-topology?
Examples, counterexamples, and partial results for conformally flat manifolds \((M,g)\) are given. This paper is related to O. Druet, E. Hebey, and M. Vaugon [Nonlinear Anal., Theory Methods Appl. 51, 79-94 (2002; Zbl 1066.35032)].
For the entire collection see [Zbl 1052.58001].

MSC:

35J60 Nonlinear elliptic equations
58J05 Elliptic equations on manifolds, general theory

Citations:

Zbl 1066.35032
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