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Tianxiang Gou

Publications and source records attributed to Tianxiang Gou.

At least 19 recordsLinked to original sources
arXiv2026

Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + ωu=|u|^{p-2}u \quad \mbox{in} \,\, \R^d, $$ where $d \geq 2$, $0 0$ and $2<p<\frac{2d}{d-2(1-a)}$. We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the natural conjectures posed recently in \cite{IS}.

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arXiv2026

NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering

We investigate a class of nonlinear equations of Schrödinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +Δu &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u \quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, \end{align*} where $N \geq 1$, $b_1, b_2>0$ and $p_1,p_2>2$. First, we establish the existence/nonexistence, symmetry, decay, uniqueness, non-degeneracy and instability of ground states. Then, we prove the scattering versus blowup below the ground state energy threshold. Our approach relies on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We also obtain an upper bound of the blow-up rate. The novelty here is that the equation does not enjoy any scaling invariance due to the presence of competing nonlinearities and the singular weights prevent the invariance by translation in the space variable. To the best of authors knowledge, this is the first time when inhomegeneous NLS equation with a focusing leading order nonlinearity and a defocusing perturbation is investigated.

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arXiv2025

Nonlinear bound states with prescribed angular momentum in the mass supercritical regime

In this paper, we consider the existence, orbital stability/instability and regularity of bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions for the mass supercritical case. Such solutions, which are given by time-dependent rotations of a non-radially symmetric spatial profile, correspond to critical points of the underlying energy function restricted on the double constraints consisting of the mass and the angular momentum. The study exhibits new pictures for rotating Bose-Einstein condensates within the framework of Gross-Pitaevskii theory. It is proved that there exist two non-radially symmetric solutions, one of which is local minimizer and the other is mountain pass type critical point of the underlying energy function restricted on the constraints. Moreover, we derive conditions that guarantee that local minimizers are regular, the set of those is orbitally stable and mountain pass type solutions are strongly unstable. The results extend and complement the recent ones in \cite{NSS}, where the consideration is undertaken in the mass subcritical case.

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arXiv2025

Non-degeneracy and uniqueness of ground states to nonlinear elliptic equations with mixed local and nonlocal operators

This paper concerns the non-degeneracy and uniqueness of ground states to the following nonlinear elliptic equation with mixed local and nonlocal operators, $$ -Δu +(-Δ)^s u + λu=|u|^{p-2}u \quad \mbox{in} \,\,\, B, \quad u=0 \quad \mbox{in} \,\,\, \R^N \backslash {B}, $$ where $N \geq 2$, $0 -λ_1$, $(-Δ)^s$ denotes the fractional Laplacian, $λ_1>0$ denotes the first Dirichlet eigenvalue of the operator $-Δ+(-Δ)^s$ in $B$ and $B$ denotes the unit ball in $\R^N$. We prove that the second eigenvalue to the linearized operator $-Δ+(-Δ)^s -(p-1)u^{p-2}$ in the space of radially symmetric functions is simple, the corresponding eigenfunction changes sign precisely once in the radial direction, where $u$ is a ground state. By deriving a new Hopf type lemma, we then get that $-λ$ cannot be an eigenvalue of the linearized operator, which in turns leads to the non-degeneracy of ground states. Moreover, by establishing a Picone type identity with respect to antisymmetric functions, we then derive the non-degeneracy of ground states in the space of non-radially symmetric functions. Relying on the non-degeneracy of ground states and adapting a blow-up argument together with a continuation argument, we then obtain the uniqueness of ground states.

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arXiv2025

Fractional Morrey-Sobolev type embeddings and nonlocal subelliptic problems with oscillating nonlinearities on stratified Lie groups

In this paper, we establish the fractional Morrey-Sobolev type embeddings on stratified Lie groups. This extends and complements the Sobolev type embeddings derived in \cite{GKR}. As an application of the results, we study the following nonlocal subelliptic problem, \begin{equation} \begin{cases} (-Δ_{\mathbb{G}, p})^s u= λβ(x) g(u) & \text{in} \quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases} \end{equation} where $0 0})$ and $g \in C(\mathbb{R}, \R) $ oscillates near the origin or at infinity. By using the variational principle of Ricceri, we prove the existence and asymptotic behaviors of infinitely many solutions to the problem under consideration. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group and $p=2$.

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arXiv2025

Radial 3D Focusing Energy Critical INLS equations with defocusing perturbation: Ground states, Scattering, and Blow-up

We investigate the following inhomogeneous nonlinear Schrödinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +Δu =|x|^{-a} |u|^{p-2} u - |x|^{-b} |u|^{4-2b}u \quad \mbox{in} \,\, \mathbb{R}_t \times \mathbb{R}_x^3, $$ where $0<a$, $b<2$ and $2+\frac{4-2a}{3}< p\leq 6-2a$. First, we establish the existence and nonexistence of ground states, along with their quantitative properties. Subsequently, we analyze the dichotomy between scattering and blow-up for solutions with energy below the ground-state energy threshold. An intriguing feature of this equation is the lack of scaling invariance, which arises from the competing effects of the inhomogeneous nonlinearities. Additionally, the presence of singular weights breaks translation invariance in the spatial variable, introducing further complexity to the analysis. To the best of our knowledge, this work represents the first comprehensive study of the inhomogeneous nonlinear Schrödinger equation with a leading-order focusing energy-critical inhomogeneous nonlinearity and a defocusing perturbation. Our results provide new insights into the interplay between these competing nonlinearities and their influence on the dynamics of solutions.

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arXiv2024

Normalized solutions to nonlinear Schrödinger equations with competing Hartree-type nonlinearities

In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree-type nonlinearities, $$ -Δu + λu=\left(|x|^{-γ_1} \ast |u|^2\right) u - \left(|x|^{-γ_2} \ast |u|^2\right) u\quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<γ_2 < γ_1 <\min\{N, 4\}$ and $λ\in \R$ appearing as Lagrange multiplier is unknown. First we establish the existence of ground states in the mass subcritical, critical and supercritical cases. Then we consider the well-posedness and dynamical behaviors of solutions to the Cauchy problem for the associated time-dependent equations.

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arXiv2024

Blowup of cylindrically symmetric solutions for biharmonic NLS

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS), $$ \textnormal{i} \, \partial_t u=Δ^2 u-μΔu-|u|^{2 σ} u \quad \text{in} \,\, \R \times \R^d, $$ where $d \geq 1$, $μ\in \R$ and $0<σ<\infty$ if $1 \leq d \leq 4$ and $0<σ<4/(d-4)$ if $d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.

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arXiv2024

Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0 -λ_{1,s}$, $2 0$ is the lowest eigenvalue of the operator $(-Δ)^s + |x|^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation.

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arXiv2024

Blow-up of cylindrically symmetric solutions for Fractional NLS

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional NLS, $$ \textnormal{i} \, \partial_t u=(-Δ)^s u-|u|^{2 σ} u \quad \text{in} \,\, \R \times \R^N, $$ where $N \geq 2$, $1/2 <s<1$ and $0<σ<2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data in \cite{BHL}.

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arXiv2024

Solutions for fourth order anisotropic nonlinear Schrödinger equations in $\R^2$

In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schrödinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_tψ+\partial_{xx} ψ-\partial_{yyyy} ψ+|ψ|^{p-2} ψ=0, \\ &ψ(0)=ψ_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schrödinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schrödinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schrödinger equations. The results are easily extendable to the higher dimensional case.

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arXiv2024

Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$, $0 -λ_{1,s}$, $2 0$ is the lowest eigenvalue of $(-Δ)^s + |x|^2$. 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. This solves an open question in \cite{SS} concerning the uniqueness of ground states.

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arXiv2024

Non-autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, $$ -Δ_p^a u-Δ_q u =λm(x) |u|^{q-2}u \quad \mbox{in} \,\, \R^N, $$ where {$N \geq 2$}, {$1<p, q<N$, $p \neq q$}, ${a \in C^{0, 1}(\R^N, [0, +\infty))}$, $a \not\equiv 0$ and $m: \R^N \to \R$ is {an indefinite sign weight which may admit nontrivial positive and negative parts}. Here $Δ_q$ is the $q$-Laplacian operator and $Δ_p^a$ is the weighted $p$-Laplace operator defined by $Δ_p^a u:=\textnormal{div}(a(x) |\nabla u|^{p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of $a$ in $\R^N$ may be zero. Our main results distinguish between the cases $p<q$ and $q<p$. In the first case, we establish the existence of a {\it continuous} family of eigenvalues, starting from the principal frequency of a suitable single phase eigenvalue problem. In the latter case, we prove the existence of a {\it discrete} family of positive eigenvalues, which diverges to infinity.

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arXiv2024

Standing waves with prescribed $L^2$-norm to nonlinear Schrödinger equations with combined inhomogeneous nonlinearities

In this paper, we are concerned with solutions to the following nonlinear Schrödinger equation with combined inhomogeneous nonlinearities, $$ -Δu + λu= μ|x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N, $$ under the $L^2$-norm constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $μ=\pm 1$, $2<q<p<{2(N-b)}/{(N-2)^+}$, $0<b<\min\{2, N\}$ and the parameter $λ\in \R$ appearing as Lagrange multiplier is unknown. In the mass subcritical case, we establish the compactness of any minimizing sequence to the minimization problem given by the underlying energy functional restricted on the constraint. As a consequence of the compactness of any minimizing sequence, orbital stability of minimizers is derived. In the mass critical and supercritical cases, we investigate the existence, radial symmetry and orbital instability of solutions. Meanwhile, we consider the existence, radial symmetry and algebraical decay of ground states to the corresponding zero mass equation with defocusing perturbation. In addition, dynamical behaviors of solutions to the Cauchy problem for the associated dispersive equation are discussed.

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arXiv2023

On the solitary waves for anisotropic nonlinear Schrödinger models on the plane

The focussing anisotropic nonlinear Schrödinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0 2$. Here the equation is of anisotropy, it means that dispersion of solutions along $x$-axis and $y$-axis is different. We show that while localized time-periodic waves, that are solutions in the form $u=e^{-\mathrm{i} ωt} ϕ$, do not exist in the regime $p\geq p_s:=\frac{2(1+s)}{1-s}$, they do exist in the complementary regime $2<p<p_s$. In fact, we construct them variationally and we establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration are easily extendable to the higher dimensional situation. We also show uniqueness of these waves under a natural weak non-degeneracy assumption. This assumption is actually removed for $s$ close to $1$, implying uniqueness for the waves in the full range of parameters.

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arXiv2022

Concentration Phenomenon of Semiclassical States to Reaction-Diffusion Systems

In this paper, we consider concentration phenomenon of semiclassical states to the following $2M$-component reaction-diffusion system in $\R \times \R^N$, \begin{align*} \left\{ \begin{aligned} \partial_t u &=\eps^2 Δ_x u-u-V(x)v + \partial_v H(u, v),\\ \partial_t v &=-\eps^2 Δ_x v+v + V(x)u - \partial_u H(u, v), \end{aligned} \right. \end{align*} where $M \geq 1$, $N \geq 1$, $\eps>0$ is a small parameter, $V \in C^1(\R^N, \, \R)$, $H \in C^1(\R^M \times \R^M, \, \R)$ and $(u, v): \R \times \R^N \to \R^M \times \R^M$. It is proved that there exist semiclassical states concentrating around the local minimum points of $V$ under mild assumptions. The approach is variational, which is mainly based upon a new linking-type argument, iterative techniques and interior estimates for nonlinear parabolic equations.

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arXiv2022

Existence and dynamics of normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians

In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians $$ (-Δ)^{s_1} u +(-Δ)^{s_2} u + λu=|u|^{p-2} u $$ under the constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0 2s_1$, $λ\in \R$ appearing as Lagrange multiplier is unknown. 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. First we establish the existence of ground state solutions and the multiplicity of bound state solutions. Then we study dynamics of solutions to the Cauchy problem for the associated time-dependent equation. Moreover, we establish orbital instability of ground state solutions.

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