Normalized solutions to lower critical Choquard equation with mixed local-nonlocal operators

This paper investigates the existence and non-existence of normalized ground state solutions for the following Choquard equation with mixed local and nonlocal operators, involving the Hardy-Littlewood-Sobolev (HLS) lower critical exponent. For the critical case $p=2+\frac{4}{N}$, we employ fibering map analysis to establish the non-existence of solutions. In the subcritical regime $2<p<2+\frac{4}{N}$, we utilize variational methods to prove the existence of normalized ground states, which are shown to be radially symmetric and strictly decreasing in $|x|$. For the supercritical case $2+\frac{4}{N}<p<2_s^{\ast }$, we introduce a homotopy-stable family to construct a Palais–Smale sequence with a negative Lagrange multiplier. By analyzing the compactness properties of this sequence, we demonstrate the existence of normalized ground state solutions in this regime as well.