Normalized solutions for Kirchhoff equations with Choquard nonlinearity: mass Super-Critical Case
Zhi-Jie Wang , Hong-Rui Sun
Communications in Analysis and Mechanics ›› 2025, Vol. 17 ›› Issue (2) : 317 -340.
In the present paper, we investigated the existence of normalized solutions for the following Kirchhoff equation with Choquard nonlinearity $-\left(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} d x\right) \Delta u-\lambda u=\mu|u|^{q-2} u+\left(I_{\alpha} *|u|^{p}\right)|u|^{p-2} u, \quad x \in \mathbb{R}^{3}$with prescribed mass $\int_{\mathbb{R}^{3}}|u|^{2} d x=c^{2}$, where a, b, c > 0, µ ∈ $\mathbb{R}$, α ∈ (0, 3), 10/3 ≤ q < 6, (3+α)/3 ≤ p < 3+α and λ ∈ $\mathbb{R}$ is a Lagrange multiplier. We first considered the case of µ > 0 and obtained mountain pass type solutions. For the defocusing situation µ < 0, we proved the existence result by constructing a minimax characterization for the energy functional. Finally, we discussed the asymptotic behavior of normalized solutions obtained above as b → 0+ when µ > 0.
Choquard nonlinearity / Kirchhoff equation / normalized solutions / defocusing case / asymptotic behavior
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