Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

Conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

“…As commented in the introduction, in this paper we will first reduce our problem to the case h = 0. In this subsection we recall some information that appears in [28] adapted to such situation. The first result is the following Kazdan-Warner identity, which is just contained in [28, Lemma 2.6 and Proposition 2.7]).…”

“…Proof. The proof is basically contained in [28]. Indeed, λ n appears in the profile description of [28, (5.4)], see also Lemma 5.2 there.…”

“…Indeed, λ n appears in the profile description of [28, (5.4)], see also Lemma 5.2 there. We recall the estimate of the term (5.7) in [28], and replace the partial derivatives of K with those of G. This leads to the term (II) in the proof of [28,Theorem 5.3]; see in particular the asymptotic expression of II 21 there. By making h = 0 we can conclude the expression given in b).…”

“…Regarding the blow-up analysis of solutions, the case K = 0 has been covered in [24], whereas in [18] a new approach is given under mild conditions on the sequence of functions h n . For the general case, a rather complete result has been recently given in [28]. We state it in a version which is adequate for our purposes (see Theorem 1.1 and Remark 1.2 in [28]).…”

“…For the general case, a rather complete result has been recently given in [28]. We state it in a version which is adequate for our purposes (see Theorem 1.1 and Remark 1.2 in [28]).…”