Existence of Solutions for Biharmonic Equations on Conical Singular Manifolds

Abstract

The manifolds with singularities are of great importance in geometric analysis. In general, the singularities include cones, edges, corners, or higher order singularities. In this paper, we study the following Dirichlet problem for a class of degenerate fourth-order elliptic equations on the conical singular manifolds.

$$\begin{aligned} \left\{ \begin{array}{l} \Delta _{{\mathbb {B}}}^2u=\lambda u+|u|^{p-2}u\quad \text{ in } \text {int}{\mathbb {B}}, \\ u=\frac{\partial u}{\partial \nu }=0 \ \ \ \quad \quad \quad \quad \quad \text {on}\ \partial {\mathbb {B}}, \end{array}\right. \end{aligned}$$

where $\lambda \ge 0$, $2<p<2^*=\frac{2n}{n-4}$, $\nu$ is the unit outward normal vector to the boundary of ${\mathbb {B}}$. $2^*=\frac{2n}{n-4}$ is the critical cone Sobolev exponent for the fourth-order problem and $n\ge 5$. We first introduce the background of conical manifolds and the weighted cone Sobolev spaces. Then we recall the cone Sobolev inequality and cone Poincaré inequality. With the help of variational method, we obtain the existence of non-trivial weak solutions in the weighted cone Sobolev space ${\mathcal {H}}_{2,0}^{2,\frac{n}{2}}({\mathbb {B}})$.