On a variational problem with lack of compactness: the topological effect of the critical points at infinity

Singular limits in Liouville-type equations

“…Another construction of these solutions, for the related mean field version of Problem (1.1) in a compact two-dimensional Riemannian manifold was carried out by Chen and Lin as a major step in their program for computation of degrees in [15]. Their construction shares elements with that of [2] but the functional-analytic setting is closer to that of [1,29] where bubbling for problems at the critical exponent was analyzed. This construction also seems to rely in essential way on the assumption that the corresponding analogue of ϕ m has only non-degenerate critical points.…”

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

“…Once v is defined by Proposition 2.2, we estimate the numbers A, B, C by taking the scalar product of (E v ) with δ 1 , P δ 2 , ∂δ 1 /∂λ 1 , ∂P δ 2 /∂λ 2 , ∂δ 1 /∂x 1 and ∂P δ 2 /∂x 2 respectively. Thus, as in [5], we get a quasi-diagonal system in the variables A, B and C i 's. The other hand side is given by …”

“…The proof follows the ideas introduced in [5]. Once v is defined by Proposition 2.2, we estimate the numbers A, B, C by taking the scalar product of (E v ) with δ 1 , P δ 2 , ∂δ 1 /∂λ 1 , ∂P δ 2 /∂λ 2 , ∂δ 1 /∂x 1 and ∂P δ 2 /∂x 2 respectively.…”

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…Conversely, Rey in [40,41] proved that any C 1 −stable critical point z0 of the Robin's function generates a family of solutions which blows-up at z0 as ǫ goes to zero. MussoPistoia in [35] and Bahri-Li-Rey in [3] studied existence of solutions which blow-up at κ different points of Ω. Grossi-Takahashi [26] proved the nonexistence of positive solutions blowing up at κ ≥ 2 points for these problems in convex domains.…”

“…In the following we describe the main steps to get some of the previous results. We will refer to [35] and [3,21,31] for the proofs related to the construction of positive and signchanging multi-bubbles to problems (BN )ǫ and (AC)ǫ, respectively. We will refer to [36] and to [33,24] for the proofs related to the construction of towers of bubbles to problems (AC)ǫ and (C)ǫ, respectively.…”