In this paper, we deal with the following (N, q)-Kirchhoff–Choquard problem with exponential growth in \({\mathbb {R}}^N\) : \(\begin{aligned} & -(1+a\int \limits _{{\mathbb {R}}^N}|\nabla u|^N\textrm{d}x)\Delta _Nu-(1+b\int \limits _{{\mathbb {R}}^N}|\nabla u|^q\textrm{d}x)\Delta _qu\\ & \qquad + V(\epsilon x)(|u|^{N-2}u+|u|^{q-2}u)\\ & \quad =[|x|^{-\mu }*F(u)]f(u), \end{aligned}\) where \(\epsilon >0\) is a small parameter, a, b are positive constants, \(1<q<N<2q\) , \(N\ge 3\) , \(0<\mu <N\) , \(\Delta _{\mathfrak {s}}u=div(|\nabla u|^{{\mathfrak {s}}-2}\nabla u)\) with \({\mathfrak {s}}\in \left\{ N,q\right\} \) is the \({\mathfrak {s}}\) -Laplacian, the nonlinear function f has an exponential growth at infinity and the potential function V is continuous in \({\mathbb {R}}^N\) . By variational methods and Lusternik–Schnirelmann category theory, we establish multiplicity and concentration of solutions for above problem as small values of the parameter \(\epsilon >0\) .
