Existence of spike-layer solutions for singularly perturbed Choquard equations

In this paper, we consider the following singularly perturbed nonlinear elliptic problem with Dirichlet boundary condition,$ \begin{equation} \nonumber \left\lbrace \begin{aligned} &-\varepsilon^{2}\Delta u+u = \varepsilon^{\mu-N}\left( \displaystyle{\int}_{\Omega}\frac{\left|u(y)\right| ^{p}}{\left| x-y\right| ^{\mu}}dy\right) \left|u\right| ^{p-2}u,\; u>0&\text {in}\; \Omega,\\ &u\left( x\right) = 0&\text {on}\; \partial \Omega. \end{aligned} \right. \end{equation} $Under certain conditions on $ p,N $, and $ \mu $, we first obtain a Liouville type result for the Choquard equation on the half space. Then, through an energy estimate, we prove that the mountain pass solution $ u_{\varepsilon} $ of the problem is a 'spike-layer' solution with a sharp spike located in the most-centered part of $ \Omega $ as $ \varepsilon $ tends to 0.