We study the existence and asymptotic behavior of normalized solutions to the following Choquard equation \(\begin{aligned} \begin{aligned}&-\Delta u + \lambda u =\mu g(u) + \gamma (I_\alpha * |u|^{\frac{N+\alpha }{N}})|u|^{\frac{N+\alpha }{N}-2}u&\text {in } \mathbb {R}^N \end{aligned} \end{aligned}\) under the \(L^2\) -norm constraint \(\int _{\mathbb {R}^N}u^2 dx =c^2\) . Here \(\gamma >0\) , \( N\ge 1\) , \(I_{\alpha }\) is the Riesz potential of order \(\alpha \in (0,N)\) , \(\mu >0\) is a parameter and the unknown \(\lambda \) appears as a Lagrange multiplier. In a mass supercritical setting on g, by establishing a novel compactness lemma and some prior energy estimate, we find regions in the \((c,\mu )\) –parameter space such that the corresponding equation admits a positive radial ground state solution, and then study the asymptotic profiles of the ground states as \((c,\mu )\) varies. In particular, we show that as \(\mu \) or c tends to 0 (resp. \(\mu \) or c tends to \(+\infty \) ), after a suitable rescaling the ground state solutions converge in \(H^1(\mathbb {R}^N)\) to a particular solution of some limit equations. Our main results are new even for the power type nonlinearity \(g(u)= |u|^{q-2}u\) with \(2+\frac{4}{N}<q<2^*\) ( \(2^*:=\frac{2N}{N-2}\) , if \(N\ge 3\) and \(2^* = \infty \) , if \(N=1, 2\) ). Further, we study the non-existence and multiplicity of positive radial solutions to \(\begin{aligned} -\Delta u + u = \eta |u|^{q-2}u + (I_\alpha * |u|^{\frac{N+\alpha }{N}})|u|^{\frac{N+\alpha }{N}-2}u, \quad \text {in}\ \ \mathbb {R}^N\end{aligned}\) where \(N \ge 1\) , \( 2+\frac{4}{N}\le q<2^*\) and \(\eta >0\) . Particularly, if \( 2+\frac{4}{N}< q<2^*\) , we show that there exist two constants \(0<\eta _1\le \eta _2<\infty \) such that the corresponding equation has a positive radial least action solution if and only if \(\eta \ge \eta _1\) and admits two positive solutions if \(\eta >\eta _2\) . To the best of our knowledge, this seems to be the first result concerning the non-existence and multiplicity of positive solutions to Choquard type equations involving the lower critical exponent.
