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Shuangjie Peng

Publications and source records attributed to Shuangjie Peng.

29 records · Page 2Linked to original sources
arXiv2017

Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity

We study the following fractional Schrödinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in C(\mathbb{R}^N;[0,\infty))$ has a local minimum and $p\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\,N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\liminf_{|x|\to \infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schrödinger; vanishing potential; penalized technique; variational methods.

math.AP↗
arXiv2017

Multi-peak positive solutions to a class of Kirchhoff equations

In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1 0$ is a parameter. Under some mild assumptions on the function $V$, we obtain multi-peak solutions for $ε$ sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{\mathbb{R}^3}|\nabla u|^2\right)Δu$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of elliptic singular perturbation problems.

math.AP↗
arXiv2017

Existence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic system

In this paper, we study the following fractional nonlinear Schrödinger system $$ \left\{% \begin{array}{ll} (-Δ)^s u +u=μ_1 |u|^{2p-2}u+β|v|^p|u|^{p-2}u,~~x\in \R^N,\vspace{2mm}\\ (-Δ)^s v +v=μ_2 |v|^{2p-2}v+β|u|^p|v|^{p-2}v,~~x\in \R^N, \end{array}% \right. $$ where $0 0, μ_2>0, 1 2s$, and $β\in \R$ is a coupling constant. We investigate the existence and non-degeneracy of proportional positive vector solutions for the above system in some ranges of $μ_1,μ_2, p, β$. We also prove that the least energy vector solutions must be proportional and unique under some additional assumptions.

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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↗
arXiv2017

Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem

We are concerned with the following Schrödinger-Newton problem \begin{equation} -\varepsilon^2Δu+V(x)u=\frac{1}{8π\varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(ξ)}{|x-ξ|}dξ\big)u,~x\in \mathbb R^3. \end{equation} For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from those of Schrödinger equations.

math.AP↗
arXiv2016

Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in $\mathbb{R}^3$

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhoff equations is found, and some related theories on classical Schrödinger equations are applied.

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arXiv2015

Existence and local uniqueness of bubbling solutions for poly-harmonic equations with critical growth

\begin{abstract} We consider the following poly-harmonic equations with critical exponents: \begin{equation}\label{P} (-Δ)^m u =K(y)u^{\frac{N+2m}{N-2m}},\;\;\; u>0\;\;\;\hbox{in} \mathbb{R}^N, \end{equation} where $N> 2m+2,m\in\mathbb{N}_{+}, K(y)$ is positive and periodic in its first $k$ variables $(y_1,\cdots, y_k)$, $1\leq k<\frac{N-2m}{2}$. Under some conditions on $K(y)$ near its critical point, we prove not only that problem~\eqref{P} admits solutions with infinitely many bubbles, but also that the bubbling solutions obtained in our existence result are locally unique. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the scalar curvature $K(y).$

math.AP↗
arXiv2014

Infinitely many positive solutions for nonlinear fractional Schrödinger equations

We consider the following nonlinear fractional Schrödinger equation $$ (-Δ)^su+u=K(|x|)u^p,\ \ u>0 \ \ \hbox{in}\ \ R^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

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arXiv2013

Segregated Vector Solutions for linearly coupled Nonlinear Schrödinger Systems

We consider the following system linearly coupled by nonlinear Schrödinger equations in $\R^3$ $$ \left\{\begin{array}{ll} -Δu_j+u_j=u^3_j-\va\sum\limits_{i\neq j}^N u_i,\{1cm}& x\in \R^3, \{0.2cm}\\ u_j\in H^1(\R^3),\quad j=1,\cdots,N, \end{array} \right. $$ where $\va\in\R$ is a coupling constant. This type of system arises in particular in models in nonlinear $N$-core fiber. We examine the effect of the linear coupling to the solution structure. When $N=2,3$, for any prescribed integer $\ell\ge 2$, we construct a non-radial vector solutions of segregated type, with two components having exactly $\ell$ positive bumps for $\va>0$ sufficiently small. We also give an explicit description on the characteristic features of the vector solutions.

math.AP↗
arXiv2013

Concentrating Bound States for Kirchhoff type problems in ${\R^3}$ involving critical Sobolev exponents

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{\R^3} {{{\left| {\nabla u} \right|}^2}} \Bigr)Δu + V(z)u = f(u) + {u^5}{\text{in}}{\R^3}, \hfill u \in {H^1}({\R^3}),{\text{}}u > 0{\text{in}}{\R^3}, \hfill \\ \end{gathered} \right.\] where $\varepsilon $ is a small positive parameter and $a,b > 0$ are constants, $f \in {C^1}({\R^ +},\R)$ is subcritical, $V:{\R^3} \to \R$ is a locally Hölder continuous function. We first prove that for ${\varepsilon_0} > 0$ sufficiently small, the above problem has a weak solution ${u_\varepsilon}$ with exponential decay at infinity. Moreover, ${u_\varepsilon}$ concentrates around a local minimum point of $V$ in $Λ$ as $\varepsilon \to 0$. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topology construct of the set where the potential $V\left(z \right)$ attains its minimum.

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