Weighted anisotropic Sobolev inequality with extremal and associated singular problems

Abstract: For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $\Omega\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_{\Omega}|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_{\Omega}\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(\Omega,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(\Omega,w)$ is the weighted 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 $$ \mu(\Omega):=\inf_{u\in W_{0}^{1,p}(\Omega,w)}\Bigg{\int_{\Omega}\mathcal{F}(\nabla u)^p w\,dx:\int_{\Omega}|u|^{q}f\,dx=1\Bigg}\leqno{\mathcal{(Q)}} $$ is associated with singular weighted anisotropic $p$-Laplace equations. To this end, we also study existence and regularity properties of solutions for weighted anisotropic $p$-Laplace equations under the mixed and exponential singularities.