For a bounded smooth domain \(\Omega \subset {\mathbb {R}}^N\) with \(N\ge 2\), we establish a weighted and an anisotropic version of Sobolev inequality related to the embedding \(W_{0}^{1,p}(\Omega )\hookrightarrow L^q(\Omega )\) for \(1<p<\infty \) and \(2\le p<\infty \) respectively. Our main emphasize is the case of \(0<q<1\) and we deal with a class of Muckenhoupt weights. Moreover, we obtain existence results for weighted and anisotropic p-Laplace equation with mixed singular nonlinearities and observe that the extremals of our inequalities are associated to such singular problems.

对于带\（N \ ge 2 \）的有界光滑域\（\ Omega \ subset {\ mathbb {R}} ^ N \），我们建立了与嵌入\（W_ {0} ^ {1，p}（\ Omega）\ hookrightarrow L ^ q（\ Omega）\）分别用于\（1 <p <\ infty \）和\（2 \ le p <\ infty \）。我们的主要重点是\（0 <q <1 \）的情况，我们处理了一类Muckenhoupt权重。此外，我们获得了具有混合奇异非线性的加权各向异性p -Laplace方程的存在性结果，并观察到我们不等式的极值与此类奇异问题有关。