Regularity and Dirichlet Problem for Double-Phase Energy Functionals of Different Power Growth

We work with a double-phase energy functional exhibiting \(\log L\) $logL$ -perturbed \( p \& q\) $p\&q$ -growth. We look for regularity properties of such functional in the setting of Musielak-Orlicz-Sobolev space, by imposing suitable conditions on the data. We further obtain the existence and uniqueness results for the solution of perturbed Dirichlet double-phase problems. We deal with both the cases of uncontrolled growth and controlled growth.