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Hong-Rui Sun

Publications and source records attributed to Hong-Rui Sun.

5 recordsLinked to original sources
arXiv2026

Existence of a Normalized Mountain Pass Solution for a Coupled Cubic Schrödinger System with Partial Confinement

This paper studies a mass supercritical Schrödinger system with partial confinement: $$ \begin{cases} -Δu_1+|y|^2u_1 =λ_1u_1+μ_1u_1^3+2βu_1u_2^2,\qquad x\in\R^3,\\ -Δu_2+|y|^2u_2 =λ_2u_2+μ_2u_2^3+2βu_1^2u_2,\qquad x\in\R^3, \end{cases} $$ subject to the mass constraints $ \int_{\R^3}u_1^2\dd x=a_1$, $\int_{\R^3}u_2^2\dd x=a_2,$ where $x=(y,z)\in\R^2\times\R$, the parameters $μ_1,μ_2,β$ are positive, and $λ_1,λ_2\in\R$ are unknown Lagrange multipliers. The masses satisfy $(a_1,a_2)=\eps(ρ_1,ρ_2)$ for a fixed vector $ρ=(ρ_1,ρ_2)\in (0,\infty)^2$. For all sufficiently small $\eps>0$, we establish the existence of a positive normalized solution $u_\eps^-$ at a mountain pass level on the full product mass sphere. This level coincides with the infimum of the energy over the negative part of the Pohozaev set. The main difficulty arises from the lack of compactness due to partial confinement. To overcome this difficulty, we develop a compactness argument that preserves the two independent mass constraints and show that every minimizing sequence for this infimum admits a strongly convergent subsequence after suitable common translations along the unconfined axial direction. We further establish the asymptotic relation $\eps E(u_\eps^-)\to m_0(ρ)$ as $\varepsilon\to0$, where $m_0(ρ)>0$ denotes the infimum of the free energy under the corresponding Pohozaev constraint. In addition, we derive the asymptotic behavior of the kinetic and potential energies. The solution obtained here is distinct from the normalized ground states constructed by Jia et al. [Discrete Contin. Dyn. Syst., 40(2020), 2739-2766] through local minimization for sufficiently small masses.

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arXiv2026

Normalized solutions to a focusing biharmonic Schrödinger equation with mixed dispersion and Sobolev critical growth

This paper investigates normalized solutions to a focusing biharmonic Schrödinger equation with mixed dispersion and combined nonlinearities involving a lower-power perturbation and a Sobolev critical term. Under a suitable smallness condition on the prescribed mass and in the range \(2<p<2+\frac{4}{N}\), we establish the existence of a local minimizer of the associated energy functional on \(S_r(c)\) and infinitely many pairs of radial normalized solutions by combining the truncation technique, the concentration-compactness principle and genus theory. In particular, our results extend those of Alves et al. (Anal. Appl., (2025), doi: 10.1142/S0219530525500447) by removing the additional assumptions imposed in their work. In the range \(2+\frac{8}{N}<p<4^*\), we prove the existence of bounded radial constrained Palais-Smale sequences at mountain pass levels below the critical Sobolev threshold for sufficiently large \(μ\). We further show that their nonzero weak limits are radial weak solutions, with strong convergence in the derivative norms and in the critical Lebesgue norm. Moreover, under an additional condition \((N)\) ensuring the strict negativity of the limiting Lagrange multiplier, the weak limit preserves the prescribed \(L^2\)-mass, and the Palais-Smale sequence converges strongly in \(H^2(\mathbb R^N)\). These results extend the previous studies of Chang et al. (arXiv preprint, (2023), arXiv: 2305.00327v1) and Bonheure et al. (Trans. Amer. Math. Soc., (2019), doi:10.1090/tran/7769).

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arXiv2026

Spectral Characterization, Local Rigidity and Morse Index of Bubble Solutions to the CR Fractional Yamabe Problem

We study bubble solutions to the critical CR fractional Yamabe equation on the Heisenberg group for the full range \(0<γ<\frac{Q}{2}\), including the admissible integer-order cases. The CR fractional operator is genuinely nonlocal and is described by Gamma-function spectral multipliers. Its structure differs from that of the Euclidean fractional Laplacian, so the linearized analysis cannot be obtained by a direct extension of integer-order arguments. We develop a direct spectral approach based on the Cayley intertwining relation and the complex spherical-harmonic decomposition, which characterizes the kernel of the linearized operator in terms of infinitesimal translation and dilation modes without reducing the equation to an integral equation or requiring a separate decay analysis. As a byproduct, we obtain the exact normalization constant of the standard bubble throughout the admissible range. We then prove three main results: $(i)$ the non-degeneracy of the bubble manifold; $(ii)$ the local rigidity of the bubble manifold, via a uniform invertibility estimate on the orthogonal complement of the tangent space, with an explicit optimal constant; $(iii)$ the Morse index of every positive bubble is exactly one. The results provide a unified spectral and variational framework for the CR Yamabe problem on the Heisenberg group, covering both integer and noninteger orders.

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arXiv2025

Existence and asymptotical behavior of normalized solutions to focusing biharmonic HLS upper critical Hartree equation with a local perturbation

This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} Δ^2u-λu-μ|u|^{p-2}u-(I_α*|u|^{4^*_α})|u|^{4^*_α-2}u=0\ \ \mbox{in}\ \mathbb{R}^N, \\[0.1cm] \int_{\mathbb{R}^N} u^2 dx = c, \end{cases} $$ where $0<α 0$, $2+\frac{8}{N}=:\bar{p}\leq p<4^*:=\frac{2N}{N-4}$, $4^*_α:=\frac{N+α}{N-4}$, $λ\in \mathbb{R}$ is a Lagrange multiplier and $I_α$ is the Riesz potential. Choosing an appropriate testing function, one can derive some reasonable estimate on the mountain pass level. Based on this point, we show the existence of normalized solutions by verifying the \emph{(PS)} condition at the corresponding mountain pass level for any $μ>0$. The contribution of this paper is that the recent results obtained for $L^2$-subcritical perturbation by Chen et al. (J. Geom. Anal. 33, 371 (2023)) is extended to the case $\bar{p}\leq p<4^*$. Moreover, we also discuss asymptotic behavior of the energy to the mountain pass solution when $μ\to 0^+$ and $c\to 0^+$, respectively.

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