arXiv subjects

Juntao Sun

Publications and source records attributed to Juntao Sun.

At least 19 recordsLinked to original sources

arXiv2026

Stabilization by a background magnetic field: global well-posedness of the full compressible viscous non-resistive MHD system without heat-conductivity

We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus $\mathbb T^3$ in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation $(\mathbf u_0,\,P_0-\bar P,\,\mathbf H_0-\mathbf n)$ of the equilibrium state $(\mathbf 0,\bar P,\mathbf n)$ is sufficiently small in a high-order Sobolev space and the background magnetic field $\mathbf n\in\mathbb R^3$ satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field $\mathbf n$, combined with a Poincar\'e-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.

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arXiv2026

Uniqueness of Ground State Solutions for a Defocusing Hartree Equation via Inverse Optimal Problems

We study a generalized defocusing Hartree equation with nonlocal exchange potential and repulsive Hartree--Fock interaction. Using an inverse optimal problem (IOP) approach, we prove the existence and uniqueness of ground state solutions. Additionally, we establish the existence of principal solutions, their continuous dependence on parameters, and a dual variational formulation. The IOP method provides a systematic framework for addressing inverse problems in nonlocal Schr\"{o}dinger operators and offers new insights into the structure of solutions for defocusing Hartree-type equations.

math.AP↗

arXiv2025

On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime

In this paper, we investigate solutions with prescribed $L^{2}$-norm (i.e., prescribed mass) for the planar Schr\"{o}dinger-Poisson (SP) equation% \begin{equation*} -\Delta u+\lambda u+\alpha \left( \log |\cdot |\ast |u|^{2}\right) u=|u|^{p-2}u,\ \text{in}\ \Omega_{R} , \end{equation*}% where $\lambda \in \mathbb{R}$ is unknown, $\alpha <0,p>4$ and $\Omega_{R} \subseteq \mathbb{R}^{2}$ is a domain. First, we prove that the energy functional $J$ corresponding to the SP equation in $\mathbb{R}^{2}$ is unbounded both above and below on the Pohozaev manifold $\mathcal{P}$; this explains the reason why the minimax level of $J$ is difficult to determine, as referenced in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Second, we establish the existence of a ground state and a high-energy solution, both with positive energy in a large bounded domain $\Omega_{R} $, which is a substantial advancement in addressing an open problem proposed in [Cingolani and Jeanjean, SIAM J. Math. Anal., 2019]. Finally, we analyze the asymptotic behavior of solutions as the domain $\Omega_{R} $ is extended to the entire space $\mathbb{R}^{2}$.

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arXiv2025

Normalized solutions for the NLS equation with potential in higher dimension: the purely Sobolev critical case

We study normalized solutions for the nonlinear Schrodinger (NLS) equation with potential and Sobolev critical nonlinearity. By establishing suitable assumptions on the potential, together with new techniques, we find a mountain-pass type solution for N>=6, which solves an open problem presented in a recent paper [Verzini and Yu, arXiv:2505.05357v1]. Moreover, we also find a local minimizer with negative energy for N>=3, which improves the results in [Verzini and Yu, arXiv:2505.05357v1].

math.AP↗

arXiv2025

Three bound states with prescribed angular momentum to the cubic-quintic NLS equations in $\mathbb{R}^{3}$

In this paper, we investigate bound states with prescribed angular momentum and mass for the nonlinear Schrödinger equations (NLS) with the cubic-quintic nonlinearity in dimensions three. We demonstrate that there exist three solutions for the double constrained problem: a local minimizer, a mountain pass type solution, and a global minimizer. Moreover, by means of the minimax method, we construct a new mountain pass path and further obtain the geometric link among the three solutions as well as a comparison of their energy levels. This seems to be the first paper concerning three solutions, with the method also being applicable to the single constraint problem.

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arXiv2025

Standing waves with prescribed mass for biharmonic NLS with positive dispersion and Sobolev critical exponent

We investigate standing waves with prescribed mass for a class of biharmonic Schrodinger equations with positive Laplacian dispersion in the Sobolev critical regime. By establishing novel energy inequalities and developing a direct minimization approach, we prove the existence of two normalized solutions for the corresponding stationary problem. The first one is a ground state with negative level, and the second one is a higher-energy solution with positive level. It is worth noting that we do not work in the space of radial functions, and do not use Palais-Smale sequences so as to avoid applying the relatively complex mini-max approach based on a strong topological argument. Finally, we explore the relationship between the ground states and the least action solutions, some asymptotic properties and dynamical behavior of solutions, such as the orbital stability and the global existence.

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arXiv2024

Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity

We investigate normalized solutions for a class of nonlinear Schrödinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+β|u|^{p-2}u$ on a bounded domain $Ω$. Firstly, when $2+\frac{4}{N}<q<p\leq2^*:=\frac{2N}{N-2}$ and $β=-1$, under an explicit smallness assumption on $V$, we prove the existence of a global minimum solution and a high-energy solution if the mass is large enough. For this case we do not require that $Ω$ is star-shaped, which partly solves an open problem by Bartsch et al. [Math. Ann. 390 (2024) 4813--4859]. Moreover, we find that the global minimizer also exists although the nonlinearity is $L^2$-supercritical. Secondly, when $2<q<2+\frac{4}{N}<p=2^*$ and $β=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $Ω$ is star-shaped and the mass is small enough. It seems to be new in the study of normalized ground state in the context of the Brézis-Nirenberg problem, even for the autonomous case of $V(x)\equiv0$.

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arXiv2024

Standing waves with prescribed mass for Schrodinger equations with competing Van Der Waals type potentials

We investigate standing waves with prescribed mass for a class of Schrodinger equations with competing Van Der Waals type potentials, arising in a model of non-relativistic bosonic atoms and molecules. By developing an approach based on a direct minimization of the energy functional on a new constrained manifold, we establish the existence of two normalized solutions for the corresponding stationary problem. One is a local minimizer with positive level and the other one is a global minimizer with negative level. Moreover, we find that the global minimizer is farther away from the origin than the local minimizer. Finally, we explore the relations between the ground state solution and the least action solution, and some dynamical behavior and scattering results are presented as well.

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arXiv2024

Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials

In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -Δu+V\left( x\right) u+ρ\left( x\right) ϕ_{ρ,\left(u,v\right) }u=\left\vert u\right\vert ^{p-2}u+β\left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}-2}u & \text{ in }\mathbb{R}^{3}, \\ -Δv+V\left( x\right) v+ρ\left( x\right) ϕ_{ρ,\left( u,v\right) }v=\left\vert v\right\vert ^{p-2}v+β\left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}-2}v & \text{ in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $ϕ_{ρ,\left( u,v\right) }=\int_{\mathbb{R}^{3}}\frac{ρ\left( y\right) \left( u^{2}(y)+v^{2}\left( y\right) \right) }{|x-y|}dy,$ the potentials $V(x),ρ(x)$ are positive continuous functions in $\mathbb{R}^{3},$ the parameter $β\in \mathbb{R}$ and $2<p<4$. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, whose main characteristic is the presence of the double coupled terms. When $2<p<3,$ under suitable assumptions on potentials, we shed some light on the behavior of the corresponding energy functional on $H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3}),$ and prove the existence of a global minimizer with negative energy. When $3\leq p<4,$ we find vectorial ground states by developing a new analytic method and exploring the conditions on potentials. Finally, we study the phenomenon of symmetry breaking of ground states when $2<p<3.$

math.AP↗

arXiv2024

Existence and dynamical behaviour of vectorial standing waves with prescribed mass for Hartree-Fock type systems

In this paper, we investigate vectorial standing waves with prescribed mass for the Hartree-Fock type system (HF system) with the double coupled feature. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics. By exploring the interaction of the double coupled terms, we prove the exis?tence/nonexistence and symmetry of vectorial energy ground states for the corresponding stationary problem. Furthermore, we obtain the relation between vectorial energy ground states and vectorial action ground states in some cases. Finally, we establish conditions for global well-posedness and finite time blow-up to HF system with the initial data, and prove orbital stability/strong instability of standing waves.

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arXiv2023

Vectorial ground state solutions for a class of Hartree-Fock type systems with the double coupled feature

In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+λϕ_{u,v}u=\left\vert u\right\vert ^{p-2}u+β\left\vert v\right\vert ^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}% -2}u & \text{ in }\mathbb{R}^{3}, \\ -Δv+v+λϕ_{u,v}v=\left\vert v\right\vert ^{p-2}v+β\left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}% -2}v & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $ϕ_{u,v}(x)=\int_{\mathbb{R}^{3}}\frac{u^{2}(y)+v^{2}\left( y\right) }{|x-y|}dy,$ the parameters $λ,β>0$ and $2<p<4$. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, taking into account the Pauli principle. Its characteristic feature lies on the presence of the double coupled terms. When $2<p<3,$ we establish the existence and multiplicity of nontrivial radial solutions, including vectorial ones, in the radial space $% H_{r}$ by describing the internal relationship between the coupling constants $λ$ and $β.$ When $2<p<4,$ we study the existence of vectorial solutions in the non-radial space $H$ by developing a novel constraint method, together with some new analysis techniques. In particular, when $3\leq p<4,$ a vectorial ground state solution is found in $% H$, which is innovative as it was not discussed at all in any previous results. Our study can be regarded as an entire supplement in d'Avenia et al. [J. Differential Equations 335 (2022) 580--614].

math.AP↗

arXiv2023

The planar Schrodinger--Poisson system with exponential critical growth: The local well-posedness and standing waves with prescribed mass

In this paper, we investigate a class of planar Schrödinger-Poisson systems with critical exponential growth. We establish conditions for the local well-posedness of the Cauchy problem in the energy space, which seems innovative as it was not discussed at all in any previous results. By introducing some new ideas and relaxing some of the classical growth assumptions on the nonlinearity, we show that such system has at least two standing waves with prescribed mass, where one is a ground state standing waves with positive energy, and the other one is a high-energy standing waves with positive energy. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable.

math.AP↗

arXiv2023

Standing waves for the NLS equation with competing nonlocal and local nonlinearities: the double $L^{2}$-supercritical case

We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stablility and strong instability of standing waves by considering a suitable local minimization problem and by analyzing the fibering mapping, respectively. A new analysis of the fibering mapping is performed in this work. We believe that it is innovative as it was not discussed at all in any previous results. The lower bound rate of blow-up solutions for the Cauchy problem is given as well. Due to the different \textquotedblleft strength" of the two types of nonlinearities, we find some essential differences in our results between these two competing cases. We will be dealing with two major scenarios that are totally different from each other due to their diverse geometric structure. This leads to surprising findings. Additionally, the competing pure power-type nonlinearities case can be derived from our study thanks to a good choice of the kernel of the Hartree term.

math.AP↗

arXiv2020

On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) Δu+μV(x)u=λf(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\geq 3,2 0$ and $μ$ sufficiently large, we obtain that at least one positive solution exists for $% 0<λ\leqλ_{1}(f_Ω) $ while at least two positive solutions exist for $λ_{1}(f_{Ω})< λ<λ_{1}(f_Ω)+δ_{a}$ without any assumption on the integral $% \int_{Ω}g(x)ϕ_{1}^{p}dx$, where $λ_{1}(f_{Ω})>0$ is the principal eigenvalue of $-Δ$ in $H_{0}^{1}(Ω)$ with weight function $f_{Ω}:=f|_{Ω}$, and $ϕ_{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2 0$ small and $0<λ<λ_{1}(f_{Ω})$; $% (ii)$ under the classical assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $λ_{1}(f_{Ω})\leq λ<λ_{1}(f_Ω)+\overline{δ}% _{a} $; $(iii)$ under the assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $λ^{+}_{a}< λ<λ_{1}(f_Ω)$ for some $a_{0}(p)>0$ and $λ^{+}_{a}\geq0$.

math.AP↗

arXiv2019

On the Kirchhoff type equations in $\mathbb{R}^{N}$

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) Δu+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2<p<\min \left\{ 4,2^{\ast }\right\}$($2^{\ast }=\infty $ for $N=1,2$ and $2^{\ast }=2N/(N-2)$ for $N\geq 3)$ and the function $f\in C(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1\leq N\leq4$ while at least two positive solutions are permitted for $N\geq5$.

math.AP↗

arXiv2018

The effect of nonlocal term on the superlinear Kirchhoff type equations in $\mathbb{R}^{N}$

We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) Δu+λV\left( x\right) u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,$ $λ>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying $\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $x\in \mathbb{R}^{N}$ for any $2<k<2^{\ast}$($2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.

math.AP↗

arXiv2018

Bound state nodal solutions for the non-autonomous Schrödinger--Poisson system in $\mathbb{R}^{3}$

In this paper, we study the existence of nodal solutions for the non-autonomous Schrödinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+λK(x) ϕu=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -Δϕ=K(x)u^{2} & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $λ>0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $λ$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $\mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.

math.AP↗

arXiv2018

Multiplicity and concentration of nontrivial solutions for the generalized extensible beam equations

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left\{ \begin{array}{ll} Δ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) Δu+λV(x) u=f( x,u) & \text{ in }\mathbb{R}^{N}, \\ u\in H^{2}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 3$, $M(t) =at^{δ}+b$ with $a,δ>0$ and $b\in \mathbb{% R}$, $λ>0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $% f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R}).$ Unlike most other papers on this problem, we allow the constant $b$ to be nonpositive, which has the physical significance. Under some suitable assumptions on $V(x)$ and $f(x,u)$, when $a$ is small and $λ$ is large enough, we prove the existence of two nontrivial solutions $u_{a,λ}^{(1)}$ and $% u_{a,λ}^{(2)}$, one of which will blow up as the nonlocal term vanishes. Moreover, $u_{a,λ}^{(1)}\rightarrow u_{\infty}^{(1)}$ and $% u_{a,λ}^{(2)}\rightarrow u_{\infty}^{(2)}$ strongly in $H^{2}(\mathbb{% R}^{N})$ as $λ\rightarrow\infty$, where $u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_{0}^{2}(Ω)$ are two nontrivial solutions of Dirichlet BVPs on the bounded domain $Ω$. It is worth noting that the regularity of weak solutions $u_{\infty}^{(i)}(i=1,2)$ here is explored. Finally, the nonexistence of nontrivial solutions is also obtained for $a$ large enough.

math.AP↗