Blow-up solutions for Hardy–Sobolev equations on compact Riemannian manifolds

Abstract

Let (M, g) be a smooth compact Riemannian manifold of dimension $n\ge 6$, $\xi _0\in M$, and we are concerned with the following Hardy–Sobolev elliptic equations:

$$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$

(0.1)

where $\Delta _g\,=\,\mathrm{div}_g(\nabla )$ is the Laplace–Beltrami operator on M, h(x) is a $C^1$ function on M, $\epsilon$ is a sufficiently small real parameter, $2^{*}(s):=\frac{2(n-s)}{n-2}$ is the critical Hardy–Sobolev exponent with $s\in (0,2)$, and $d_{g}$ is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).