arXiv subjects

Lipeng Duan

Publications and source records attributed to Lipeng Duan.

8 recordsLinked to original sources

arXiv2026

Solutions with one dimensional concentration for a two dimensional Gross-Pitaevskii model with general potential

We concern standing wave solutions with frequency $\lambda$ to a two dimensional Gross-Pitaevskii equation with a trap potential under the unit mass constraint, which is used to describe Bose-Einstein condensates with attractive interaction. First, we investigate the necessary conditions for existence of the solutions with concentration phenomena directed along closed smooth curves. Next, not only imposing stationary and non-degeneracy conditions on the curves with respect to an auxiliary weighted length involving the trap potential, but also adding some other technical assumptions, we select a sequence $\{\lambda_j\}$ of the frequency $\lambda$ with $-\lambda_j\rightarrow +\infty$ and construct solutions with concentration directed along the curves. Our result partially answers the conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Comm. Math. Phys. 2003] about necessary condition for solution concentrating at submanifolds. The solutions constructed in this paper are concentrating on curves whose length are non-uniformly bounded, and hence the situation is quite different from that in [M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007].

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arXiv2022

The helical vortex filaments of Ginzburg-Landau system in ${\mathbb R}^3$

We consider the following coupled Ginzburg-Landau system in ${\mathbb R}^3$ \begin{align*} \begin{cases} -ε^2 Δw^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -ε^2 Δw^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^-=0, \end{cases} \end{align*} where $w=(w^+, w^-)\in \mathbb{C}^2$ and the constant coefficients satisfy $$ A_+, A_->0,\quad B^2 0, \quad {t^+}^2+{ t^-}^2=1. $$ If $B<0$, then for every $ε$ small enough, we construct a family of entire solutions $w_ε(\tilde{z}, t)\in \mathbb{C}^2$ in the cylindrical coordinates $(\tilde{z}, t)\in \mathbb{R}^2 \times \mathbb{R}$ for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in {\tt arXiv:1901.02807}. These solutions are $2π$-periodic in $t$ and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in $\mathbb{R}^3$ which has been studied in {\tt arXiv:1901.02807}. Our results negatively answer the Gibbons conjecture \cite{Gibbons conjecture} for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.

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arXiv2022

Doubling the equatorial for the prescribed scalar curvature problem on ${\mathbb{S}}^N$

We consider the prescribed scalar curvature problem on $ {\mathbb{S}}^N $ $$ Δ_{{\mathbb S}^N} v-\frac{N(N-2)}{2} v+\tilde{K}(y) v^{\frac{N+2}{N-2}}=0 \quad \mbox{on} \ {\mathbb S}^N, \qquad v >0 \quad \mbox{on} \ {\mathbb S}^N, $$ under the assumptions that the scalar curvature $\tilde K$ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of $O(3)$ obtained doubling the equatorial. We use the finite dimensional Lyapunov-Schmidt reduction method.

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arXiv2022

Positive solutions for a critical elliptic equation

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -Δu= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~Ω},\\[1mm] u>0,~ &{\text{in}~Ω},\\[1mm] u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\geq 3$, $s\in [1,2^*-1)$ with $2^*=\frac{2N}{N-2}$, $\varepsilon>0$, $Ω$ is a smooth bounded domain in $\mathbb{R}^N$. Under some conditions on $Q(x)$, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997, 461--483) proved that there exists a single-peak solution for small $\varepsilon$ if $N\geq 4$ and $s\in (1,2^*-1)$. And they proposed in Remark 1.7 of their paper that \vskip 0.1cm\begin{center} \emph{``it is interesting to know the existence of single-peak solutions for small $\varepsilon$ and $s=1$''.} \end{center}\vskip 0.1cm \noindent Also it was addressed in Remark 1.8 of their paper that \vskip 0.1cm \begin{center} \emph{``the question of solutions concentrated at several points at the same time is still open''.} \end{center}\vskip 0.1cm \noindent Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether $s=1$ or $s>1$.

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arXiv2021

Semi-classical analysis for Fractional Schrödinger Equations with fast decaying potenials

We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} ε^{2s}(-Δ)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and $V(x)$ is a nonnegative continuous potential. We use penalized technique to show that the problem has a family of solutions concentrating at a positive local minimum of $V(x)$ provided that $\frac{2s}{N-2s}+2<p<\frac{2N}{N-2s}$. The novelty is that $V$ can decay arbitrarily or even be compactly supported.

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arXiv2020

New type of solutions for the Nonlinear Schrödinger Equation in $\mathbb{R}^N$

We construct a new family of entire solutions for the nonlinear Schrödinger equation \begin{align*} \begin{cases} -Δu+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases} \end{align*} where $p\in (1, \frac{N+2}{N-2})$ and $N\geq 3$, and $V (y)= V(|y|)$ is a positive bounded radial potential satisfying $$ V(|y|) = V_0 + \frac{a}{|y|^m} + O( \frac{1}{|y|^{m+σ}} ), \quad {\mbox {as}} \quad |y| \to \infty , $$ for some fixed constants $V_0, a, σ>0$, and $m>1$. Our solutions have strong analogies with the doubling construction of entire finite energy sign-changing solution for the Yamabe equation.

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arXiv2020

Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain

We consider the inhomogeneous Allen-Cahn equation $$ ε^2Δu\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ Ω, \qquad \frac {\partial u}{\partial ν}\,=\,0\quad \mbox{on}\ \partial Ω, $$ where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary $\partialΩ$ and $V(x)$ is a positive smooth function, $ε>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. For any fixed integer $N\geq 2$, we will show the existence of a clustered solution $u_ε$ with $N$-transition layers near $\partial Ω$ with mutual distance $O(ε|\ln ε|)$, provided that the generalized mean curvature $\mathcal{H} $ of $\partialΩ$ is positive and $ε$ stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by A. Malchiodi, W.-M. Ni, J. Wei in Pacific J. Math. (Vol. 229, 2007, no. 2, 447-468) and A. Malchiodi, J. Wei in J. Fixed Point Theory Appl. (Vol. 1, 2007, no. 2, 305-336)

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arXiv2019

On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system

For the following Ginzburg-Landau system in ${\mathbb R}^2$ \begin{align*} \begin{cases} -Δw^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -Δw^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^-=0, \end{cases} \end{align*} with constraints $ A_+, A_->0$, $B<0$, $B^2 0$, we will concern its linearized operator ${\mathcal L}$ around the radially symmetric solution $w(x)=(w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2$ of degree pair $(1, 1)$ and prove the non-degeneracy result: the kernel of ${\mathcal L}$ is spanned by $\big\{\frac{\partial w}{\partial{x_1}}, \frac{\partial w}{\partial{x_2}}\big\}$ in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator ${\mathcal L}$ will be given.

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