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Yuxia Guo

Publications and source records attributed to Yuxia Guo.

At least 19 recordsLinked to original sources
arXiv2026

Topological degree and existence of nontrivial normalized solutions for mass-critical Gross-Pitaevskii systems

In this paper, we study the existence of nontrivial solutions for the following Gross-Pitaevskii system involving mass-critical exponent: \[ \left\{ \begin{array}{ll} -Δu_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+βu_{1}u_{2}^2+μu_{1}& \hbox{ in }Ω,\\ -Δu_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+βu_{2}u_{1}^2+μu_{2}&\hbox{ in }Ω, u_{1},u_{2}\ge 0 &\hbox{ in }Ω, u_1=u_2=0 &\hbox{ on }\partialΩ, \end{array}\right. \] with the constraint \[ \int_Ω(u_1^2+u_2^2)=1, \] where $Ω$ is an unbounded smooth domain in $\mathbb{R}^2$, $a_1, a_2, β$ are positive parameters, $V_i$ are trapping potentials, and $μ\in\mathbb{R}$ is an unknown Lagrange multiplier. We derive the existence of solutions by computing the Leray-Schauder degree for the parameters $a_1,a_2, β$, which are away from some critical values. The system may have semi-trivial solutions of the form $(u_1, 0)$ or $(0, u_2)$. Our novelty is that we provide mechanisms ensuring that the solutions we find are nontrivial, i.e., $u_1>0$ and $u_2>0$.

math.AP↗
arXiv2026

Rigidity of weak solutions for anisotropic N-Laplacian equation with Neumann or Robin boundary condition

This paper is devoted to the rigidity of weak solutions for anisotropic $N$-Laplacian equations with Neumann or Robin boundary conditions on smooth bounded convex domains of $\mathbb{R}^N$. The anisotropic operator is given by $$a(ξ) = H^{N-1}(ξ)\nabla H(ξ),$$ where $H$ stands for a norm on $\mathbb{R}^N$; this formulation contains the classical $N$-Laplacian as a special case. We establish a key integral inequality involving the anisotropic gradient and the second fundamental form of the domain boundary, which acts as the core technical tool in our proofs. Under natural monotonicity assumptions on the nonlinearity, we prove that all weak solutions to the Neumann boundary problem are constant, without requiring any a priori boundedness assumption on the solution. Furthermore, we extend this rigidity result to Robin boundary value problems by imposing suitable constraints on the boundary nonlinear term. Moreover, our rigidity results remain valid not only on bounded convex domains but also on suitable unbounded domains. By working under substantially weaker assumptions than those previously available, we establish rigidity results that fill the gaps in the existing literature for anisotropic $N$-Laplacian equations with nonlinear boundary conditions and substantially extend the rigidity theory of anisotropic quasilinear elliptic equations at the critical exponent $p=N$.

math.AP↗
arXiv2026

Infinitely many sign-changing solutions for critical Hamiltonian systems with linear perturbation

In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -Δu = |v|^{p-1} v + ε(αu + β_1 v), & \text{in } Ω, \\ -Δv = |u|^{q-1} u + ε(β_2 u + αv), & \text{in } Ω, \\ u = v = 0, & \text{on } \partial Ω, \end{cases} \tag{*} \end{equation} where \(Ω\) is the unit ball in $\mathbb{R}^N$, \(ε\) is a small parameter, \(α\), \(β_1\) and \(β_2\) are real numbers, \((p, q)\) is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.

math.AP↗
arXiv2025

Construction of solutions for a critical elliptic system of Hamiltonian type

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -Δu + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -Δv + V(|y'|,y'')\, v = |u|^{q-1}u, & \text{in } \mathbb{R}^N, \end{cases} \end{equation*} where $(y', y'') \in \mathbb{R}^2 \times \mathbb{R}^{N-2}$, $V(|y'|, y'') \not\equiv 0$ is a bounded, nonnegative function on $\mathbb{R}_+ \times \mathbb{R}^{N-2}$ and $p, q > 1$ lie on the critical hyperbola: \[ \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}. \] By applying the finite-dimensional reduction method and local Pohozaev identities combined with the Green representation formula and technical analysis, we show that, under the assumptions that $N \ge 5$, $(p,q)$ lies in a certain admissible range, and $r^2 V(r, y'')$ has a stable critical point, the above problem admits infinitely many solutions whose energy can be made arbitrarily large.

math.AP↗
arXiv2025

Sign-changing solutions for critical Hamiltonian systems in $\mathbb{R}^N$

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -Δu =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -Δv =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the exponents $(p,q)$ satisfy $p,q>1$ and belong to the critical hyperbola $$\frac1{p+1}+\frac1{q+1} =\frac {N-2}N.$$ To establish this result, we introduce several new ideas and strategies that are both robust and potentially applicable to other critical problems lacking the Kelvin invariance.

math.AP↗
arXiv2024

Non-degeneracy and new type of cylindrial solutions for a critical Grushin-type problem

In this paper, we consider a critical Grushin-type problem, which is closely related to the prescribed Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature. We first prove a non-degeneracy result through local Pohozaev identities, then by using the Lyapunov-Schmidt reduction methods, we construct new type of multi-bubbling solutions with cylindrical symmetry.

math.AP↗
arXiv2024

Non-degeneracy of solution for critical Lane-Emden systems with linear perturbation

In this paper, we consider the following elliptic system \begin{equation*} \begin{cases} -Δu = |v|^{p-1}v +ε(αu + β_1 v), &\hbox{ in }Ω, \\-Δv = |u|^{q-1}u+ε(β_2 u +αv), &\hbox{ in }Ω, \\u=v=0,&\hbox{ on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, $ε$ is a small parameter, $α$, $ β_1$ and $ β_2$ are real numbers, $(p,q)$ is a pair of positive numbers lying on the critical hyperbola \begin{equation*} \begin{split} \frac{1}{p+1}+\frac{1}{q+1} =\frac{N-2}{N}. \end{split} \end{equation*} We first revisited the blowing-up solutions constructed in \cite{Kim-Pis} and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions.

math.AP↗
arXiv2024

Multiple Boundary Peak Solution for Critical Elliptic System with Neumann Boundary

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -Δu + μu=v^p, &\hbox{in } Ω, \\-Δv + μv=u^q, &\hbox{in } Ω, \\\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, &\hbox{on } \partialΩ, \\u>0,v>0, &\hbox{in } Ω, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $μ$ is a positive constant and $(p,q)$ lies in the critical hyperbola: $$ \dfrac{1}{p+1} + \dfrac{1}{q+1} =\dfrac{N-2}{N}. $$ By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary $\partial Ω$. Our results show that the geometry of the boundary $\partialΩ,$ especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.

math.AP↗
arXiv2024

New type of solutions for Schrödinger equations with critical growth

We consider the following nonlinear Schrödinger equations with critical growth: \begin{equation} - Δu + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial function in $C^1$, $N\ge 5$. By using a finite reduction argument, we show that if $r^2V(r)$ has either an isolated local maximum or an isolated minimum at $r_0>0$ with $V(r_0)>0$, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of $O(3)$.

math.AP↗
arXiv2023

Non-degeneracy of double-tower solutions for nonlinear Schrödinger equation and applications

This paper is concerned with the following nonlinear Schrödinger equation \begin{equation} \label{eq} - Δu + V(|y|)u=u^{p},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \ \ \ u \in H^1(\mathbb {R}^N), \end{equation} where $V(|y|)$ is a positive function, $1<p <\frac{N+2}{N-2}$. Based on the local Pohozaev identities and blow-up analysis, we first prove a non-degeneracy result for double-tower solutions constructed in [18] in a suitable symmetric space. As an application, we obtain the existence of new type solutions for (0.1).

math.AP↗
arXiv2023

Double-tower Solutions for Higher Order Prescribed Curvature Problem

We consider the following higher order prescribed curvature problem on $ {\mathbb{S}}^N : $ \begin{equation*} D^m \tilde u=\widetilde{K}(y) \tilde u^{m^{*}-1} \quad \mbox{on} \ {\mathbb {S}}^N, \qquad \tilde u >0 \quad \mbox{in} \ {\mathbb {S}}^N. \end{equation*} where $\widetilde{K}(y)>0$ is a radial function, $m^{*}=\frac{2N}{N-2m}$ and $D^m$ is $2m$ order differential operator given by \begin{equation*} D^m=\prod_{i=1}^m\left(-Δ_g+\frac{1}{4}(N-2i)(N+2i-2)\right), \end{equation*} where $g=g_{\mathbb{S}^N}$is the Riemannian metric. We prove the existence of infinitely many double-tower type solutions, which are invariant under some non-trivial sub-groups of $O(3),$ and their energy can be made arbitrarily large.

math.AP↗
arXiv2021

Non-degeneracy and existence of new solutions for the Schrödinger equations

We consider the following nonlinear problem $$ (P) \quad \quad - Δu + V(|y|)u=u^{p},\quad u>0 \quad \mbox{in} \ {\mathbb{R}}^N, \quad u \in H^1({\mathbb{R}}^N), $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that the multi-bump solutions constructed in [20] is non-degenerate in a suitable symmetric space. We also use this non-degenerate result to construct new solutions for (P).

math.AP↗
arXiv2019

{Localized nodal solutions for $p-$Laplacian equations with critical exponents in $\mathbb{R}^N$

In this paper, we consider the existence of localized sign-changing solutions for the $p-$Laplacian nonlinear Schrödinger equation $$ -ε^pΔ_pu+V(x)|u|^{p-2}u=|u|^{p^*-2}u+μ|u|^{q-2}u,~~u\in W^{1,p}(\mathbb{R}^N), $$ where $1 0$, $Δ_p$ is the $p-$Laplacian operator. By using the penalization method together with the truncation method and a blow-up argument, we establish for small $ε$ the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function.

math.AP↗
arXiv2019

Solutions for fractional operator problem via local Pohozaev identities

We consider the following fractional Schrödinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-Δ)^s u+V(|y'|,y'')u=u^{2^*_s-1} \ \hbox{ in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. \end{equation*} where $s\in(\frac{1}{2}, 1)$, $(y',y'')\in \mathbb{R}^2\times \mathbb{R}^{N-2}$, $V(|y'|,y'')$ is a bounded nonnegative function with a weaker symmetry condition. We prove the existence of infinitely many solutions for the above problem by a finite dimensional reduction method combining various Pohazaev identies.

math.AP↗
arXiv2018

Infinitely many non-radial solutions to a critical equation on annulus

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem: \begin{equation*} \left\{\begin{array}{rlll} -Δu&=|u|^{\frac{4}{N-2}}u, &\hbox{ in }Ω,\\ u&=0, &\hbox{ on }\partialΩ. \end{array}\right. \eqno(P) \end{equation*} on the annulus $Ω:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $N\geq 3.$ In particular, for any integer $k$ large enough, we build a non-radial solution which look like the unique positive solution $u_0$ to $(P)$ crowned by $k$ negative bubbles arranged on a regular polygon with radius $r_0$ such that $r_0^{\frac{N-2}{2}}u_0(r_0)=:\displaystyle\max_{a\leq r\leq b}r^{\frac{N-2}{2}}u_0(r).$

math.AP↗
arXiv2017

Solutions for biharmonic equations with steep potential wells

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$Δ^2 u+(λV(x)-δ)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2 0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<δ<μ_0$ and $μ_0$ is the principle eigenvalue of $Δ^2$ in the zero sets $\mbox{int} V^{-1}(0)$ of $V(x)$. Here $\mbox{int} V^{-1}(0)$ denotes the interior part of the set $V^{-1}(0):=\{x\in \mathbb{R}^N: V(x)=0\}$. We prove that the above equation admits a least energy solution which is trapped near the zero sets $\mbox{int} V^{-1}(0)$ for $λ>0$ large.

math.AP↗
arXiv2017

Uniqueness of planar vortex patch in incompressible steady flow

We investigate a steady planar flow of an ideal fluid in a bounded simple connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

math.AP↗