Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form $$\left(\int_E |u(x)|^q\,\omega(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,\sigma(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}$$ where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $\omega,\sigma:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, \omega, \sigma) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $\omega$ and $\sigma$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.