The existence and concentration behavior of positive ground state solutions for a class of Choquard type equations involving local and nonlocal operators

In the present paper, we consider a class of Choquard type equations driven by mixed local and nonlocal operators

$ \begin{align} \nonumber -\varepsilon^{2}\Delta u+\varepsilon^{2\kappa}(-\Delta)^{s}u+Vu = (I_{\alpha}*|u|^p)|u|^{p-2}u, \; \; \; x\in\mathbb{R}^N, \end{align} $

where $ N>2 $, $ s\in(0, 1) $, $ \kappa\in(s, 1) $, and $ \frac{1}{p}\in(\frac{N-2}{N+\alpha}, \frac{N}{N+\alpha}) $, $ \alpha\in(0, N) $. Under certain assumptions on nonnegative potential $ V(x) $, by variational method we show the existence of a positive ground state solution for $ \varepsilon>0 $ small enough, and investigate the concentration behavior of solutions as $ \varepsilon\rightarrow0 $ with $ \kappa $ assigned a fixed value, which depends on the homogeneity of $ V(x) $.