PRESCRIBING MEAN CURVATURE ON 𝔹ⁿ

Abstract

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball B n , n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β ∈ (n − 2, n − 1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler-Hopf type formula.

ON THE PRESCRIBED MEAN CURVATURE PROBLEM ON THE STANDARD n-DIMENSIONAL BALL

“…Roughly, it is assumed that there exists a real number β such that in some geodesic normal coordinate system centered at y, we have In [1], the authors proved that under condition (f ) β , with n − 2 < β < n − 1, (P ) has a solution provided (1.1)…”

Blowing up solutions for Neumann problem with critical nonlinearity on the boundary

“…In [11], Escobar has studied this problem on manifolds which are not equivalent to the standard ball. In the case of the ball, under some conditions on H , existence results are obtained (see [2][3][4]8,9,12]) and the references therein.…”

Sign‐changing bubble for Neumann problem with critical nonlinearity on the boundary

“…In [1], Escobar has studied this problem on manifolds that are not equivalent to the standard ball. In the case of the ball, under some conditions on H, existence results are obtained (see [2][3][4][5][6][7] and the references therein).…”