Multi-bump solutions for the double phase critical Schrödinger equations involving logarithmic nonlinearity

In this paper, we consider the existence and multiplicity of multi-bump solutions for the following double phase critical Schrödinger equations with logarithmic nonlinearity in $\mathbb{R}^N$ \begin{equation}\label{E} \begin{cases} -\Delta_pu-\Delta_qu+(\lambda\mathcal {V}(x) +\mathcal {Z}(x))(|u|^{p-2}u+|u|^{q-2}u) \\ \qquad\qquad\qquad\qquad\qquad\qquad =\alpha |u|^{p-2}u\log |u|^p+|u|^{p^\ast-2}u, \\ u\in W^{1,p}(\mathbb{R}^N)\bigcap W^{1,q} (\mathbb{R}^N), \end{cases} \tag{$\mathcal E$} \end{equation} where $N\geq3, \alpha\in (1, \infty), \Delta_\wp u=$div$(|\nabla|^{\wp-2}\nabla u)$ is the $\wp$-Laplacian operator with $\wp\in\{p,q\}$, $0 < q < p < p^\ast,$ the parameter $\lambda\geq1$ and $ \mathcal {Z},\, \mathcal {V}: \mathbb{R}^N \rightarrow \mathbb{R}$ are nonnegative continuous functions. Applying variational methods, we prove {the} existence of at least $2^k-1$ multi-bump solutions of \eqref{E} as $\lambda\geq1$ is sufficiently large. The main features and novelty of the paper {are the presence of the double phase operator and of the} logarithmic nonlinearity. Moreover, these results are new even in the case $p = q =2$. In a sense, we fill in the blanks of previous papers in the subject.