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.
Categories: math.AP