New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak-Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of $W^{1,\mathcal{H}}(\mathrm{\Omega })$ into $L^{{\mathcal{H}}_{\ast }}(\mathrm{\Omega })$, where ${\mathcal{H}}_{\ast }$ is the Sobolev conjugate function of $\mathcal{H}$, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.