In this paper, we are concerned with the existence and multiplicity of multi-bump solutions for the following (N, q)-Laplacian equation \(\begin{aligned} & -\Delta _N u-\Delta _q u+(\mu \mathcal {V}(x)+\mathcal {Z}(x))(|u|^{N-2}u+|u|^{q-2}u)\\ & \quad =h(u)+|u|^{q-2}u\log |u|^q\quad \text {in}\ \mathbb {R}^N, \end{aligned}\) where \(2\le N<q\) , \(\mu \in [1,+\infty )\) , \(\Delta _\mathfrak {s}u=\text {div}(|\nabla u|^{\mathfrak {s}-2}\nabla u)\) is the \(\mathfrak {s}\) -Laplace operator with \(\mathfrak {s}\in \{N,q\}\) , h is a continuous function with exponential critical growth, \(\mathcal {V}\) is a nonnegative continuous function with the potential well \(\Omega :=\text {int}(\mathcal {V}^{-1}(0))\) consisting of k components, and the nonnegative continuous function \(\mathcal {Z}\) verifies some assumptions. With the aid of variational methods, we obtain the existence and multiplicity of multi-bump solutions as \(\mu \) is large enough. As far as we know, it is the first time that the existence and multiplicity of multi-bump solutions to the (N, q)-Laplacian equation with exponential critical growth and logarithmic nonlinearity are studied. The most obvious and important feature is that we establish some new technique results to prove our results.
