Normalized solutions to lower critical Choquard equation in mass-supercritical setting

We study the normalized solutions to the following Choquard equation \begin{equation*} \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 \endaligned \end{equation*} under the $L^2$-norm constraint $\|u\|_2=c$. Here $\gamma>0$, $N\geq 1$, $\alpha\in(0,N)$, $I_{\alpha}$ is the Riesz potential, and the unknown $\lambda$ appears as a Lagrange multiplier. In a mass supercritical setting on $g$, we find regions in the $(c,\mu)$--parameter space such that the corresponding equation admits a positive radial ground state solution. To overcome the lack of compactness resulting from the nonlocal term, we present a novel compactness lemma and some prior energy estimate. These results are even new for the power type nonlinearity $g(u)= |u|^{q-2}u$ with $2+\frac{4}{N}<q<2^*$ ($2^*:=\frac{2N}{N-2}$, if $N\geq 3$ and $2^* = \infty$, if $N=1, 2$). We also 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 $H^1(\RN)$ to a particular solution of the limit equations. Further, we study the non-existence and multiplicity of positive radial solutions to \begin{equation*} -\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}\ \ \RN \end{equation*} where $N \geq 1$, $2< q<2^*$ and $\eta>0$. Based on some analytical ideas the limit behaviors of the normalized solutions, we verify some threshold regions of $\eta$ such that the corresponding equation has no positive least action solution or admits multiple positive solutions. 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.