Spike vector solutions for some coupled nonlinear Schrödinger equations
Shuangjie Peng Huirong Pi
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 2205-2227 doi: 10.3934/dcds.2016.36.2205
We consider spike vector solutions for the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\ -\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\ u, v >0 \,\ \hbox{in}\ \mathbb{R}^3, \end{array} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ are positive potentials, $\mu>0, \nu>0$ are positive constants and $\beta\neq 0$ is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer $k\ge 2$, we construct $k$ interacting spikes concentrating near the local maximum point $x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in the attractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ interacting spikes for $u$ near the local maximum point $x_{0}$ of $P(x)$ and $m$ interacting spikes for $v$ near the local maximum point $\bar{x}_{0}$ of $Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover, spikes of $u$ and $v$ repel each other. Meanwhile, we prove the attractive phenomenon for $\beta < 0$ and the repulsive phenomenon for $\beta > 0$.
keywords: spike vector solutions critical point. asymptotic behavior Coupled nonlinear Schrödinger equations Lyapunov-Schmidt reduction
On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling
Dengfeng Lü Shuangjie Peng
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3327-3352 doi: 10.3934/dcds.2017141

In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms $f$ and $g$, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter $β$ tends to zero.

keywords: Fractional Laplacian system Berestycki-Lions type conditions vector ground state solution asymptotic behavior Pohozaev manifold
Remarks on singular critical growth elliptic equations
Shuangjie Peng
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 707-719 doi: 10.3934/dcds.2006.14.707
Let $\Omega$ be a bounded domain in $\mathbb R^N$$(N\geq 4)$ with smooth boundary $\partial \Omega$ and the origin $0 \in \overline{\Omega}$, $\mu<0$, 2*=2N/(N-2). We obtain existence results of positive and sign-changing solutions to Dirichlet problem $-\Delta u=\mu\frac{ u}{|x|^2}$+|u|2*-2u+$\lambda u \ \text{on}\ \Omega,\ u=0 \ \text{on}\ \partial\Omega$, which also gives a positive answer to the open problem proposed by A. Ferrero and F. Gazzola in [Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177(2001), 494-522].
keywords: Hardy inequality singular elliptic equation. critical Sobolev exponents Positive and sign-changing solutions compactness
Concentration of solutions for a Paneitz type problem
Shuangjie Peng Jing Zhou
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 1055-1072 doi: 10.3934/dcds.2010.26.1055
By variational methods, we construct infinitely many concentration solutions for a type of Paneitz problem under the condition that the Paneitz curvature has a sequence of strictly local maximum points moving to infinity.
keywords: critical point concentrating solutions. variational method Paneitz type problem
Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
Wei Long Shuangjie Peng Jing Yang
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 917-939 doi: 10.3934/dcds.2016.36.917
We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
keywords: Fractional Laplacian reduction method. nonlinear scalar field equation
Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2012, 32(3): 795-826 doi: 10.3934/dcds.2012.32.795
In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
keywords: variational methods. critical exponent Multiple solutions
Infinitely many radial solutions to elliptic systems involving critical exponents
Yinbin Deng Shuangjie Peng Li Wang
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 461-475 doi: 10.3934/dcds.2014.34.461
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
keywords: elliptic systems Radial solution (PS) condition. critical exponent

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