On the prescribed scalar curvature on 3-half spheres: Multiplicity results and Morse inequalities at infinity. (English) Zbl 1159.58007

This paper deals with a generalization of the scalar curvature problem on closed Riemann manifolds. The authors are interested in the existence of a smooth positive solution in the case of a nonlinear boundary condition. The main analytic difficulties of the problem considered in the present paper are due to the presence of critical exponents. In such a case, since the Sobolev embeddings are not compact, the associated energy functional fails to satisfy the Palais-Smale compactness condition. By means of a Morse-type lemma at infinity, the authors argue that noncompact orbits of the gradient can be treated as usual critical points. It is also established that the noncompactness of the flow lines of the gradient flow does occur only if the manifold is conformally equivalent to the standard half-sphere, at least in the three dimensional case.
The proofs strongly rely on the critical point theory at infinity, introduced and developed by Abbas Bahri.


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J65 Nonlinear boundary value problems for linear elliptic equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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