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Maoding Zhen

Publications and source records attributed to Maoding Zhen.

7 recordsLinked to original sources
arXiv2021

Normalized solutions for Schr\"{o}dinger system with subcritical Sobolev exponent and combined nonlinearities

In this paper, we look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta v+\lambda_{2}v=\alpha_{2}|v|^{p-2}v+\mu_{2}v^{3}+\rho u^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the additional conditions $\int_{\mathbb{R}^{N}}u^{2}dx=b^{2}_{1}$ and $\int_{\mathbb{R}^{N}}v^{2}dx=b^{2}_{2}.$ Here $b_1, b_2>0$ are prescribed, $N\leq3$, $\mu_{1}, \mu_{2}, \alpha_{1},\alpha_{2},\rho>0$, $p\in (2,4)$ and the frequencies $\lambda_{1},\lambda_{2}$ are unknown and will appear as Lagrange multipliers. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres, we prove the existence of normalized ground states under suitable conditions on $b_1$ and $b_2$, which are obtained as global minimizers. When $N=3$, we show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. We also shows the limit behavior of the normalized solutions as $\alpha_{1},\alpha_{2}\rightarrow 0$. The first solution will disappear and the second solution will converge to the normalized solution of system (1.1) with $\alpha_{1}=\alpha_{2}=0$, which has been studied by T. Bartsch, L. Jeanjean and N. Soave (J. Math. Pures Appl. 2016). Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground states. The results in this paper complement the main results established by X. Luo, X. Yang and W. Zou (arXiv:2107.08708), where the authors considered the case $N=4$.

math.AP↗
arXiv2021

Normalized solutions for Schr\"{o}dinger system with quadratic and cubic interactions

In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schr\"{o}dinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on $b_1$ and $b_2$, we prove the existence of normalized solutions. When $N=3$, the energy functional is always unbounded on the product of $L^2$-spheres. We show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as $\beta\rightarrow 0$. Finally, we deal with the high dimensional cases $N\geq 4$. Several non-existence results are obtained if $\beta<0$. When $N=4$, $\beta>0$, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case $\beta=0$, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schr\"{o}dinger system but also leads to a stabilization of the related evolution system.

math.AP↗
arXiv2021

Normalized ground states for the critical fractional NLS equation with a perturbation

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: \begin{equation*} \begin{cases} (-\Delta)^{s}u=\lambda u +\mu|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u,&x\in\mathbb{R}^{N}, \int_{\mathbb{R}^{N}}u^{2}dx=a^{2},\\ \end{cases} \end{equation*} where $(-\Delta)^{s}$ is the fractional Laplacian, $0 2s$, $2 0$, $\mu\in \mathbb{R}$. By using Jeanjean's trick in \cite{Jeanjean}, and the standard method which can be found in \cite{Brezis} to overcome the lack of compactness, we first prove several existence and nonexistence results for a $L^{2}$-subcritical (or $L^{2}$-critical or $L^{2}$-supercritical) perturbation $\mu|u|^{q-2}u$, then we give some results about the behavior of the ground state obtained above as $\mu\rightarrow 0^{+}$. Our results extend and improve the existing ones in several directions.

math.AP↗
arXiv2018

Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent \begin{equation*} \begin{cases} (-\Delta)^{s}u+\mu u=|u|^{p-1}u+\lambda v & x\in \ \mathbb{R}^{N}, (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u& x\in \ \mathbb{R}^{N},\\ \end{cases} \end{equation*} where $(-\Delta)^{s}$ is the fractional Laplacian, $0 2s, \ \lambda <\sqrt{\mu\nu },\ 1 \mu_{0}$, there exists a $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$ such that if $\lambda>\lambda_{\mu,\nu}$, the system has a positive ground state solution, if $\lambda<\lambda_{\mu,\nu}$, the system has no ground state solution.

math.AP↗
arXiv2017

Carleman and observability estimates for stochastic beam equation

In this paper, we establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate.

math.OC↗
arXiv2017

Critical system involving fractional Laplacian

In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n}, u,v\in D_{s}(\mathbb{R}^{n}). \end{cases} \end{equation*} By using the Nehari\ manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.

math.AP↗