MULTIPLICITY AND CONCENTRATION OF SOLUTIONS TO A SINGULAR CHOQUARD EQUATION WITH CRITICAL SOBOLEV EXPONENT

In this paper, we consider a nonautonomous singular Choquard equation with critical exponent <p class="disp_formula">$ \left \{\begin{array}{lcl} -\Delta u+V(x)u+\lambda(I_\alpha\ast |u|^{p})|u|^{p-2}u=f(x)u^{-\gamma}+|u|^{4}u, &amp;&amp;\quad x\in\mathbb{R}^3,\\ u&gt;0,&amp;&amp;\quad x\in\mathbb{R}^3, \end{array}\right. $ where $ I_\alpha $ is the Riesz potential of order $ \alpha\in(0,3) $ and $ 1+\frac{\alpha}{3}\le p&lt;3 $, $ 0&lt;\gamma&lt;1 $. Under certain assumptions on $ V $ and $ f $, we show the existence and multiplicity of positive solutions for $ \lambda&gt;0 $ by using variational method and Nehari type constraint. We also study concentration of solutions as $ \lambda\rightarrow0^+ $.