arXiv subjects

Wenming Zou

Publications and source records attributed to Wenming Zou.

At least 55 records · Page 3Linked to original sources

arXiv2022

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schrödinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-δu+λ_1 u=|u|^{2^*-2}u+{να} |u|^{α-2}|v|^βu,\quad \text{in }\mathbb{R}^N, &-δv+λ_2 v=|v|^{2^*-2}v+{νβ} |u|^α|v|^{β-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $α,β>1$, $2<α+β<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $ν<0$. When $ν>0$ and $α+β\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<ν<ν_0$, the constant $ν_0$ will be explicitly given. In the case $α+β>2+\frac{4}{N}$ we prove the existence of a threshold $ν_1\ge 0$ such that a normalized ground state solution exists for $ν>ν_1$, and does not exist for $ν<ν_1$. We also give conditions for $ν_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $ν\to0^+$ or $ν\to+\infty$.

math.AP↗

arXiv2022

Least energy positive soultions for $d$-coupled Schrödinger systems with critical exponent in dimension three

In the present paper, we consider the coupled Schrödinger systems with critical exponent: \begin{equation*} \begin{cases} -Δu_i+λ_{i}u_i=\sum\limits_{j=1}^{d} β_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } Ω,\\ u_i \in H_0^1(Ω) ,\quad i= 1,2,...,d. \end{cases} \end{equation*} Here, $Ω\subset \mathbb{R}^{3}$ is a smooth bounded domain, $d \geq 2$, $β_{ii}>0$ for every $i$, and $β_{ij}=β_{ji}$ for $i \neq j$. We study a Brézis-Nirenberg type problem: $-λ_{1}(Ω)<λ_{1},\cdots,λ_{d}<-λ^*(Ω)$, where $λ_{1}(Ω)$ is the first eigenvalue of $-Δ$ with Dirichlet boundary conditions and $λ^*(Ω)\in (0, λ_1(Ω))$. We acquire the existence of least energy positive solutions to this system for weakly cooperative case ($β_{ij}>0$ small) and for purely competitive case ($β_{ij}\leq 0$) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case $N\geq 5$. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three.

math.AP↗

arXiv2021

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square-rectangular transition lattice shapes in many physical and biological system (such as Bose-Einstein condensates and two-component Ginzburg-Landau systems). It turns out, our results also apply to locating the minimizers of sum of two Eisenstein series, which is new in number theory.

math.CA↗

arXiv2021

Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: the higher dimensional case

Let $Ω\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with $d\geq 2$ equations \begin{equation*} -Δu_{i}+λ_{i}u_{i}=|u_{i}|^{p-2}u_{i}\sum_{j = 1}^{d}β_{ij}|u_{j}|^{p} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d, \end{equation*} in the case of a critical exponent $2p=2^*=\frac{2N}{N-2}$ in high dimensions $N\geq 5$. We treat the focusing case ($β_{ii}>0$ for every $i$) in the variational setting $β_{ij}=β_{ji}$ for every $i\neq j$, dealing with a Brézis-Nirenberg type problem: $-λ_{1}(Ω)<λ_{i}<0$, where $λ_{1}(Ω)$ is the first eigenvalue of $(-Δ,H^1_0(Ω))$. We provide several sufficient conditions on the coefficients $β_{ij}$ that ensure the existence of least energy positive solutions; these include the situations of pure cooperation ($β_{ij}> 0$ for every $i\neq j$), pure competition ($β_{ij}\leq 0$ for every $i\neq j$) and coexistence of both cooperation and competition coefficients. Some proofs depend heavily on the fact that $1 0$, $-λ_1(Ω)<λ<0$ for all $N\geq 4$, a result which is new in dimensions $N=4,5$.

math.AP↗

arXiv2021

A new deduce of the strict binding inequality and its application: Ground state normalized solution to Schrödinger equations with potential

In the present paper, we prove the existence of solutions $(λ, u)\in \R\times H^1(\R^N)$ to the following elliptic equations with potential $\displaystyle -Δu+(V(x)+λ)u=g(u)\;\hbox{in}\;\R^N, $ satisfying the normalization constraint $\displaystyle \int_{\R^N}u^2=a>0,$ which is deduced by searching for solitary wave solution to the time-dependent nonlinear Schrödinger equations. Besides the importance in the applications, not negligible reasons of our interest for such problems with potential $V(x)$ are their stimulating and challenging mathematical difficulties. We develop an interesting way basing on iteration and give a new proof of the so-called "sub-additive inequality", which can simply the standard process in the traditional sense. Under some very relax assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a>0$.

math.AP↗

arXiv2021

Normalized solutions for nonlinear Schrödinger systems with special mass-mixed terms: The linear couple case

In this paper, we prove the existence of positive solutions $(λ_1,λ_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schrödinger system $$\begin{cases} -Δu + λ_1 u= μ_1|u|^{p-2}u+βv \quad &\hbox{in}\;\RN, \\ -Δv + λ_2 v= μ_2|v|^{q-2}v+βu \quad &\hbox{in}\;\RN, \end{cases}$$ satisfying the normalization constraints $\displaystyle\int_{\RN}u^2 =a, ~ \int_{\RN}v^2 =b$. The parameters $μ_1,μ_2,β>0$ are prescribed and the masses $a,b>0$. Here $2+\frac{4}{N} 0$ and $β>0$, provided $2\leqslant N\leqslant 4$. For the Sobolev critical case with $N=3,4$, it can be viewed as a counterpart of the Brezis-Nirenberg critical semilinear elliptic problem for the system case in the context of normalized solutions. Under some suitable assumptions, we obtain the existence or non-existence of positive normalized ground state solution.

math.AP↗

arXiv2021

Positive normalized solutions to nonlinear elliptic systems in $\R^4$ with critical Sobolev exponent

In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -Δu+λ_1u=μ_1u^3+α_1|u|^{p-2}u+βv^2u\quad&\hbox{in}~\R^4,\\ -Δv+λ_2v=μ_2v^3+α_2|v|^{p-2}v+βu^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $μ_1,μ_2,β>0$; $α_1,α_2\in \R$, $p\!\in\! (2,4)$ and $λ_1,λ_2\!\in\!\R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $α_i<0(i=1,2)$. Then turning to the case of $α_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schrödinger systems with Sobolev critical exponent. When $3\leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.

math.AP↗

arXiv2021

Positive least energy solutions for $k$-coupled Schrödinger system with critical exponent: the higher dimension and cooperative case

In this paper, we study the following $k$-coupled nonlinear Schrödinger system with Sobolev critical exponent: \begin{equation*} \left\{ \begin{aligned} -Δu_i & +λ_iu_i =μ_i u_i^{2^*-1}+\sum_{j=1,j\ne i}^{k} β_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;Ω,\newline u_i&>0 \quad \hbox{in}\; Ω\quad \hbox{and}\quad u_i=0 \quad \hbox{on}\;\partialΩ, \quad i=1,2,\cdots, k. \end{aligned} \right. \end{equation*} Here $Ω\subset \mathbb{R}^N $ is a smooth bounded domain, $2^{*}=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-λ_1(Ω)<λ_i<0, μ_i>0$ and $ β_{ij}=β_{ji}\ne 0$, where $λ_1(Ω)$ is the first eigenvalue of $-Δ$ with the Dirichlet boundary condition. We characterize the positive least energy solution of the $k$-coupled system for the purely cooperative case $β_{ij}>0$, in higher dimension $N\ge 5$. Since the $k$-coupled case is much more delicated, we shall introduce the idea of induction. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $k$-coupled system decreases as $k$ grows. Moreover, we establish the existence of positive least energy solution of the limit system in $\mathbb{R}^N$, as well as classification results.

math.AP↗

arXiv2021

Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions

In the present paper, we study the normalized solutions for the following quasilinear Schrödinger equations: $$-Δu-uΔu^2+λu=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$ We first consider the mass-supercritical case $p>4+\frac{4}{N}$, which has not been studied before. By using a perturbation method, we succeed to prove the existence of ground state normalized solutions, and by applying the index theory, we obtain the existence of infinitely many normalized solutions. Then we turn to study the mass-critical case, i.e., $p=4+\frac{4}{N}$, and obtain some new existence results. Moreover, we also observe a concentration behavior of the ground state solutions.

math.AP↗

arXiv2021

Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -Δu+λ_1u=μ_1 |u|^{p-2}u+βr_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -Δv+λ_2v=μ_2 |v|^{q-2}v+βr_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $β$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.

math.AP↗

arXiv2020

Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation $$ (-Δ)^σu = |x|^αu^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $σ\in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2σ< α< 2σ$ and $\frac{n+α}{n-2σ} < p < \frac{n+2σ}{n-2σ}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981), but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on $\mathbb{S}^{n}_+$. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.

math.AP↗

arXiv2020

Normalized solutions for a coupled Schrödinger system

In the present paper, we prove the existence of solutions $(λ_1,λ_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schrödinger equations $$ \begin{cases} -Δu+λ_1u=μ_1 u^3+βuv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -Δv+λ_2v=μ_2 v^3+βu^2v\quad&\hbox{in}\;\mathbb{R}^3\\ u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $μ_1,μ_2,β>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $λ_1,λ_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $β$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $μ_1=μ_2$ we prove that normalized solutions exist for all $β>0$ and all $a,b>0$.

math.AP↗

arXiv2018

Qualitative analysis for an elliptic system in the punctured space

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$ \begin{cases} -Δu =μ_1 u^{2q+1} + βu^q v^{q+1} \\ -Δv =μ_2 v^{2q+1} + βv^q u^{q+1} \end{cases} \textmd{in} ~\mathbb{R}^n \backslash \{0\}, $$ where $μ_1, μ_2$ and $β$ are all positive constants, $n\geq 3$. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.

math.AP↗

arXiv2018

Exact Asymptotic Behavior of Singular Positive Solutions of Fractional Semi-Linear Elliptic Equations

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-Δ)^σu = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $σ\in (0, 1)$ and $\frac{n}{n-2σ} < p < \frac{n+2σ}{n-2σ}$.

math.AP↗

arXiv2018

On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-Δ)^σu = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).

math.AP↗

arXiv2018

Spikes of the two-component elliptic system in $\bbr^4$ with Sobolev critical exponent

Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2Δu_1+λ_1u_1=μ_1u_1^3+α_1u_1^{p-1}+βu_2^2u_1\quad&\text{in}Ω,\\ &-\ve^2Δu_2+λ_2u_2=μ_2u_2^3+α_2u_2^{p-1}+βu_1^2u_2\quad&\text{in}Ω,\\ &u_1,u_2>0\quad\text{in}Ω,\quad u_1=u_2=0\quad\text{on}\partialΩ,\endaligned\right. \end{equation*} where $Ω\subset\bbr^4$ is a bounded domain, $λ_i,μ_i,α_i>0(i=1,2)$ and $β\not=0$ are constants, $\ve>0$ is a small parameter and $2 0$ small enough. The concentration behavior of the ground state solution as $\ve\to0^+$ is also studied. Furthermore, by combining the elliptic estimates and local energy estimates, we also obtain the location of the spikes as $\ve\to0^+$. To the best of our knowledge, this is the first attempt devoted to the spikes in the Bose-Einstein condensate in $\bbr^4$.

math.AP↗

arXiv2018

On the existence and regularity of vector solutions for quasilinear systems with linear coupling

We study a class of linearly coupled system of quasilinear equations. Under some assumptions on the nonlinear terms, we establish some results about the existence and regularity of vector solutions for the p-Laplacian systems by using variational methods. In particular, we get two pairs of nontrivial solutions. We also study their different asymptotic behavior of solutions as the coupling parameter tends to zero.

math.AP↗

arXiv2017

A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\mathbb{R}^N$

In this paper, we will study the following PDE in $\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \begin{cases} Δu+\sum_{i=1}^{l}λ_i \frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\;\hbox{in}\;\mathbb{R}^N, u\in D_{0}^{1,2}(\mathbb{R}^N), \end{cases} $$ where $0 0$ for $1\leq i\leq k$; $λ_i<0$ for $k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and non-existence of the positive ground state solution. The regularity of the least-energy solution are also investigated.

math.AP↗