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.
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Bal, K., Garain, P. Weighted and anisotropic Sobolev inequality with extremal. manuscripta math. 168, 101–117 (2022). https://doi.org/10.1007/s00229-021-01298-3
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DOI: https://doi.org/10.1007/s00229-021-01298-3