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For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and abounded smooth domain $\Omega\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, weestablish the following weighted anisotropic Sobolev inequality $$S\left(\int_{\Omega}|u|^qf\,dx\right)^\frac{1}{q}\leq\left(\int_{\Omega}\mathcal{F}(\nabla u)^pw\,dx\right)^\frac{1}{p},\quad\forall\,u\inW_0^{1,p}(\Omega,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(\Omega,w)$ is theweighted Sobolev space under a class of $p$-admissible weights $w$, where $f$is some nonnegative integrable function in $\Omega$. We discuss the case$0<q<1$ and observe that $$ \mu(\Omega):=\inf_{u\inW_{0}^{1,p}(\Omega,w)}\Bigg\{\int_{\Omega}\mathcal{F}(\nabla u)^pw\,dx:\int_{\Omega}|u|^{q}f\,dx=1\Bigg\}\leqno{\mathcal{(Q)}} $$ is associatedwith singular weighted anisotropic $p$-Laplace equations. To this end, we alsostudy existence and regularity properties of solutions for weighted anisotropic$p$-Laplace equations under the mixed and exponential singularities.

Weighted anisotropic Sobolev inequality with extremal and associated singular problems