Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$

In this paper, we establish continuous and compact embeddings for a new class of Musielak-Orlicz Sobolev spaces in unbounded domains driven by a double phase operator with variable exponents that depend on the unknown solution and its gradient. Using these embeddings and an abstract critical point theorem, we prove the existence and multiplicity of weak solutions for such problems associated with this new operator in the whole space $\mathbb{R}^d$. This work can be seen as a continuation of the recent paper by Bahrouni--Bahrouni--Missaoui--R\u{a}dulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.