Abstract
In this paper we study the Nirenberg problem on standard half spheres (mathbb {S}^n_+,g), , n ge 5, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: (P)-Δgu+n(n-2)4u=Kun+2n-2,u>0inS+n,∂u∂ν=0on∂S+n.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\mathcal {P}) \\quad {\\left\\{ \\begin{array}{ll} -\\Delta _{g} u \\, + \\, \\frac{n(n-2)}{4} u \\, = K \\, u^{\\frac{n+2}{n-2}},\\, u > 0 &{}\\quad \\text{ in } \\mathbb {S}^n_+, \\\\ \\frac{\\partial u}{\\partial \\nu }\\, =\\, 0 &{}\\quad \\text{ on } \\partial \\mathbb {S}^n_+. \\end{array}\\right. } \\end{aligned}$$\\end{document}where K in C^3(mathbb {S}^n_+) is a positive function. This problem has a variational structure but the related Euler–Lagrange functional J_K lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional J_K in their neighborhood and compute their contribution to the difference of topology between the level sets of J_K, hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for (mathcal {P}) on half spheres of dimension n ge 5.
Highlights
Where K ∈ C3(Sn+) is a positive function
Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional JK in their neighborhood and compute their contribution to the difference of topology between the level sets of JK, extending the full Morse theoretical approach to this non compact variational problem
It can be proved that finite energy blowing up solutions of (N Pε) can have only isolated simple blow up points which are critical points of the function K, see [23,31,32,35]
Summary
For the function K and its restriction on the boundary K1 := K ∂Sn , we use the following assumption: (H1): We assume that K is a C3(Sn+) positive function, which has only non-degenerate critical points with K = 0. A fact which was used in [1] to construct subcritical solutions having non simple blow ups To rule out such a possibility, under our assumption (H 2) and (H 3), we had to come up with a barycentric vector field which moves a cluster of concentration points towards their common barycenter and to prove that along the flow lines of such a vector field the functional decreases and the concentration rates of an initial value do not increase, see Lemma 3.9. We collect in the appendix some estimates of the bubble, fine asymptotic expansion of the Euler–Lagrange functional and its gradient in the neighborhood at infinity as well as useful counting index formula for the critical points of the function K and its restriction K1 on the boundary
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
