中文摘要：

建立了含到边界距离的Hardy-Poincaré不等式，并得到新空间中的嵌入紧性结果．此外，考虑一类含到边界距离的半线性椭圆型方程．首先，研究相应的特征值问题并得到特征值的一些性质．然后，利用这些结果及临界点理论在一个新的Hilbert空间中证明了方程非平凡解的存在性．

英文摘要：

A Hardy-Poincaré inequality with distance to boundary δΩ was established. A compactness result of embedding was obtained in a new space. Moreover, a class of semilinear elliptic equations with distance to boundary was studied. Firstly, the corresponding eigenvalue problem was studied and some properties of eigenvalues were obtained. By means of these preliminaries and the critical point theory, several existence results of nontrivial solutions to the original equation were proved in a new Hilbert space.