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^* $, 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.
Citation: Chun Qin, Jie Yang. Normalized solutions to lower critical Choquard equation with mixed local-nonlocal operators[J]. AIMS Mathematics, 2025, 10(12): 28668-28688. doi: 10.3934/math.20251262
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^* $, 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.
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Chun Qin, Jie Yang. Normalized solutions to lower critical Choquard equation with mixed local-nonlocal operators[J]. AIMS Mathematics, 2025, 10(12): 28668-28688. doi: 10.3934/math.20251262
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