Weighted and anisotropic Sobolev inequality with extremal

Abstract

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.