Blowing-up solutions for a slightly subcritical Choquard equation

In this paper, we study the following asymptotically critical Choquard equation 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\displaystyle \left( \int _{\Omega }\frac{u^{2_\alpha ^*-\varepsilon }(y)}{|x-y|^\alpha }dy\right) u^{2_\alpha ^*-1-\varepsilon }, \ \ & \text{ in }\ \Omega ,\\ u>0,\ \ & \text{ in }\ \Omega ,\\ u=0, \ \ & \text{ on }\ \partial \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\mathrm{\Delta }u=\left({\int }_{\mathrm{\Omega }},\frac{u^{2_{\alpha }^{\ast }-\epsilon }(y)}{{|x-y|}^{\alpha }},d,y\right)u^{2_{\alpha }^{\ast }-1-\epsilon },}\hfill & \text{in}\mathrm{\Omega },\hfill \\ u>0,\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain in \(\mathbb {R}^N\) ${\mathbb{R}}^N$ , \(N\ge 3\) $N\ge 3$ , \(\alpha \in (0,N)\) $\alpha \in (0,N)$ , \(2_\alpha ^*=\frac{2N-\alpha }{N-2}\) $2_{\alpha }^{\ast }=\frac{2N-\alpha }{N-2}$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and \(\varepsilon >0\) $\epsilon >0$ is a small parameter. We prove that any family of solutions to (0.1), under a suitable constrain on their gradients, blows up exactly at a critical point of the Robin function. Conversely, by a reduction argument, we construct a family of solutions to (0.1) concentrating around a critical point of the Robin function as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ .