Non-compactness and infinite number of conformal initial data sets in high dimensions. (English) Zbl 1331.58027
On a closed compact Riemannian manifold of dimension \(n\geq 6,\) the authors investigate non-compact issues for the set of positive solutions to the Einstein-Lichnerowicz equation:
\[
\Delta_gu+hu=fu^{2^*-1}+\dfrac{a}{u^{2^*+1}},
\]
where \(h,f,a\) are given functions on \(M\) such that \(\Delta_g+h\) is coercive, \(f>0,\) \(a\geq 0\) with \(a\not\equiv0\) and \(2^*=\frac{2n}{n-2}.\) Examples of background physical coefficients are constructed for which the equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B44 | Blow-up in context of PDEs |
| 35B09 | Positive solutions to PDEs |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 35Q76 | Einstein equations |
Keywords:
blow-up solutions; finite-dimensional reduction; initial-value problem in general relativity; positive solutions; Einstein-Lichnerowicz equationReferences:
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