Sobolev and Trudinger inequalities in weighted Morrey spaces for double phase functionals

Abstract

Our aim in this paper is to establish Sobolev and Trudinger inequalities for Sobolev functions in weighted Morrey spaces. As an application, we extend these inequalities for double phase functionals $\Phi(x,t) = t^p + (b(x) t)^q$, where $1\<p\<q$ and $b(\cdot)$ is nonnegative, bounded and Hölder continuous of order $\theta \in (0,1]$. Our results are new even for the unweighted case.