A class of double phase variable exponent energy functionals with different power growth and logarithmic perturbation

In this paper, we study a new kind of double phase variational exponents energy functional with log $ L $-perturbed term, determined by the operator $ \begin{align} u\mapsto\Delta_{\mathcal {H}_{L}} u = \text{div} \left(\frac{\mathcal {H}'_{L}(x, |\nabla u|)}{|\nabla u|}\nabla u \right), \end{align} $ where$ \begin{align} \mathcal {H}_{L}(x, t) = [t^{p(x)}+\mu(x)t^{q(x)}]\log(e+\alpha t)\mbox{ for all }~~ x\in \Omega , \mbox{all} ~~ t\ge 0 , \mbox{some }~~ \alpha \geq 0 . \end{align} $The features of the related Musielak-Orlicz Sobolev spaces are delivered. We further prove the density of smooth functions in such spaces in the case when the domain is bounded, or unbounded but with certain requirements. Finally, under very general assumptions on data, we show the existence and uniqueness results for weak solutions to a special class of perturbed Dirichlet double phase problems.