ABSTRACT

In this paper, we consider the following quasilinear equation: ${\mathrm{\Delta }}_{p,g}u+a(x)|u{|}^{p-2}u=\frac{K(x)|u{|}^{p^{\ast }(s)-2}u}{d_g(x,x_0{)}^s}+h(x)|u{|}^{r-2}u,x\in M,$ where M is a compact Riemannian manifold with dimension $n⩾3$ without boundary, and $x_0\in M$. Here $a(x)$, $K(x)$ and $h(x)$ are continuous functions on M satisfying some further conditions. The operator ${\mathrm{\Delta }}_{p,g}$ is the p-Laplace–Beltrami operator on M associated with the metric g, and $d_g$ is the Riemannian distance on $(M,g)$. Moreover, we assume $p\in (1,n)$, $s\in [0,p)$, and $r\in (p,p^{\ast })$ with $p^{\ast }=\frac{np}{n-p}$. The notion $p^{\ast }(s)=\frac{(n-s)p}{n-p}$ is the critical Hardy–Sobolev exponent. With the help of Mountain Pass Theorem, we get the existence results under different assumptions.

摘要

在本文中，我们考虑以下拟线性方程：${\mathrm{\Delta }}_{p,G}你+\mathrm{一个}(X)|你{|}^{p-2}你=\frac{ķ(X)|你{|}^{p^{*}(s)-2}你}{d_G(X,X_0{)}^s}+H(X)|你{|}^{r-2}你,X\in 米,$其中M是紧黎曼流形，维数$n⩾3$没有边界，并且$X_0\in 米$. 这里$\mathrm{一个}(X)$,$ķ(X)$和$H(X)$是M上满足一些进一步条件的连续函数。运营商${\mathrm{\Delta }}_{p,G}$是与度量g相关联的M上的p -Laplace–Beltrami 算子，并且$d_G$是黎曼距离$(米,G)$. 此外，我们假设$p\in (1,n)$,$s\in [0,p)$， 和$r\in (p,p^{*})$和$p^{*}=\frac{np}{n-p}$. 观念$p^{*}(s)=\frac{(n-s)p}{n-p}$是临界 Hardy-Sobolev指数。借助山口定理，我们得到了不同假设下的存在性结果。