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Tianxiang Gou

Publications and source records attributed to Tianxiang Gou.

27 records · Page 2Linked to original sources
arXiv2022

Higher topological type semiclassical states for Sobolev critical Dirac equations with degenerate potential

In this paper, we are concerned with semiclassical states to the following Sobolev critical Dirac equation with degenerate potential, \begin{align*} -\textnormal{i} \eps α\cdot \nabla u + a βu + V(x) u=|u|^{q-2} u + |u| u \quad \mbox{in} \,\, \R^3, \end{align*} where $u:\mathbb{R}^3\rightarrow \mathbb{C}^4$, $2 0$ is a small parameter, $a>0$ is a constant, $α=(α_1, α_2, α_3)$, $α_j$ and $β$ are $4 \times 4$ Pauli-Dirac matrices. We construct an infinite sequence of higher topological type semiclassical states with higher energies concentrating around the local minimum points of the degenerate potential $V$. The solutions are obtained from a minimax characterization of higher dimensional symmetric linking structure, which correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. Our approach is variational, which mainly relies on penalization method and blow-up arguments along with local type Pohozaev identity.

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arXiv2021

Higher topological type semiclassical states for fractional nonlinear elliptic equations

In this paper, we are concerned with semiclassical states to the following fractional nonlinear elliptic equation, \begin{align*} \eps^{2s}(-Δ)^s u + V(x) u=\mathcal{N}(|u|)u \quad \mbox{in} \,\,\, \R^N, \end{align*} where $0 0$ is a small parameter, $N>2s$, $V \in C^1(\R^N, \R^+)$ and $\mathcal{N}\in C(\R, \R)$. The nonlinearity has Sobolev subcritical, critical or supercritical growth. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^N$, where $\mathcal{F}$ denotes the Fourier transform. We construct positive semiclassical states and an infinite sequence of sign-changing semiclassical states with higher energies clustering near the local minimum points of the potential $V$. The solutions are of higher topological type, which are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. They correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. The proofs are mainly based on penalization methods, s-harmonic extension theories and blow-up arguments along with local type Pohozaev identities.

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arXiv2020

Normalized solutions to the Chern-Simons-Schrödinger system

In this paper, we study normalized solutions to the Chern-Simons-Schrödinger system, which is a gauge-covariant nonlinear Schöridnger system with a long-range electromagnetic field, arising in nonrelativistic quantum mechanics theory. The solutions correspond to critical points of the underlying energy functional subject to the $L^2$-norm constraint. Our research covers several aspects. Firstly, in the mass subcritical case, we establish the compactness of any minimizing sequence to the associated global minimization problem. As a by-product of the compactness of any minimizing sequence, the orbital stability of the set of minimizers to the problem is achieved. In addition, we discuss the radial symmetry and uniqueness of minimizer to the problem. Secondly, in the mass critical case, we investigate the existence and nonexistence of normalized solution. Finally, in the mass supercritical case, we prove the existence of ground state and infinitely many radially symmetric solutions. Moreover, the instability of ground states is explored

math.AP↗
arXiv2019

Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement

We are concerned with the existence of solutions to the following nonlinear Schrödinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -Δu_1 + (x_1^2+x_2^2)u_1&= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + βr_1|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -Δu_2 + (x_1^2+x_2^2)u_2&= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 +βr_2 |u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \end{equation*} under the constraint \begin{align*} \int_{\mathbb{R}^3}|u_1|^2 \, dx = a_1>0,\quad \int_{\mathbb{R}^3}|u_2|^2 \, dx = a_2>0, \end{align*} where $μ_1, μ_2, β>0, 2 1, r_1 + r_2 < \frac{10}{3}$. In the system, the parameters $λ_1, λ_2 \in \R$ are unknown and appear as the associated Lagrange multipliers. Our solutions are achieved as global minimizers of the underlying energy functional subject to the constraint. Our purpose is to establish the compactness of any minimizing sequence up to translations. As a by-product, we obtain the orbital stability of the set of global minimizers.

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arXiv2018

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $$ γΔ^2 u -Δu + αu=|u|^{2 σ} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $γ>0, N \geq 1, 4 \leq σN < \frac{4N}{(N-4)^+}$, whereas the parameter $α\in \R$ will appear as a Lagrange multiplier. Given $c \in \R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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arXiv2018

Multiple positive normalized solutions for nonlinear Schrödinger systems

We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + βr_1 |u_1|^{r_1-2} u_1|u_2|^{r_2}, -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + βr_2 |u_1|^{r_1} |u_2|^{r_2 -2} u_2, \end{aligned} \right. \] under the constraint \[ \int_{\mathbb{R}^N}|u_1|^2 \, dx = a_1,\quad \int_{\mathbb{R}^N}|u_2|^2 \, dx = a_2. \] Here $a_1, a_2 >0$ are prescribed, $μ_1, μ_2, β>0$, and the frequencies $λ_1, λ_2$ are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when $N \geq 1, 2 < p_1, p_2 < 2 + \frac 4N, r_1, r_2 > 1, 2 + \frac 4N < r_1 + r_2 < 2^*$, the second when $ N \geq 1, 2+ \frac 4N < p_1, p_2 < 2^*, r_1, r_2 > 1, r_1 + r_2 < 2 + \frac 4N.$ In both cases, assuming that $β>0$ is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

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arXiv2017

Strong instability of ground states to a fourth order Schrödinger equation

In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.

math.AP↗
arXiv2016

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

In this paper we investigate the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for nonlinear Schrödinger systems of the form \[ \left\{ \begin{aligned} -Δu_1 &= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + r_1β|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -Δu_2 &= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 + r_2 β|u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned} \right. \] under the constraints \[\int_{R^N}|u_1|^2 \, dx = a_1>0,\quad \int_{R^N}|u_2|^2 \, dx = a_2>0. \] Here $ N \geq 1, β>0, μ_i >0, r_i >1, 2 <p_i < 2 + \frac{4}{N}$ for $i=1,2$ and $ r_1 + r_2 < 2 + \frac{4}{N}$. This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system $λ_1$ and $λ_2$ are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation, and as a consequence we obtain the orbital stability of the standing waves associated to the set of minimizers.

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