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 $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ $(N\ge 2)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(\hat{I}\text{c◯},w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $\hat{I}\text{c◯}$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0}^{1,p}(Ω,w)}\Bigg\{\int_Ω\mathcal{F}(\nabla u)^p w\,dx:\int_Ω|u|^{q}f\,dx=1\Bigg\}\leqno{\mathcal{(Q)}} $$ 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.