# American Institute of Mathematical Sciences

## Journals

DCDS
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 $\beta > 0$.
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DCDS
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.

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DCDS
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$$(N\geq 4) with smooth boundary \partial \Omega and the origin 0 \in \overline{\Omega}, \mu<0, 2*=2N/(N-2). We obtain existence results of positive and sign-changing solutions to Dirichlet problem -\Delta u=\mu\frac{ u}{|x|^2}+|u|2*-2u+\lambda u \ \text{on}\ \Omega,\ u=0 \ \text{on}\ \partial\Omega, which also gives a positive answer to the open problem proposed by A. Ferrero and F. Gazzola in [Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177(2001), 494-522]. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2010, 26(3): 1055-1072 doi: 10.3934/dcds.2010.26.1055 By variational methods, we construct infinitely many concentration solutions for a type of Paneitz problem under the condition that the Paneitz curvature has a sequence of strictly local maximum points moving to infinity. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2016, 36(2): 917-939 doi: 10.3934/dcds.2016.36.917 We consider the following nonlinear fractional scalar field equation$$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$where K(|x|) is a positive radial function, N\ge 2, 0 < s < 1, and 1 < p < \frac{N+2s}{N-2s}. Under various asymptotic assumptions on K(x) at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large. keywords: DCDS Discrete & Continuous Dynamical Systems - A 2012, 32(3): 795-826 doi: 10.3934/dcds.2012.32.795 In this paper, we consider the following problem$$ \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) $$where \lambda>0 is a parameter, 2^* =\frac {2N}{N-2} is the critical Sobolev exponent and N>4, f(x,t) and h(x) are some given functions. We prove that there exists 0<\lambda^{*}<+\infty such that (\star) has exactly two positive solutions for \lambda\in(0,\lambda^{*}) by Barrier method and Mountain Pass Lemma and no positive solutions for \lambda >\lambda^*. Moreover, if \lambda=\lambda^*, (\star) has a unique solution (\lambda^{*}, u_{\lambda^{*}}), which means that (\lambda^{*}, u_{\lambda^{*}}) is a turning point in H^{1}(\mathbb{R}^{N}) for problem (\star). keywords: DCDS Discrete & Continuous Dynamical Systems - A 2014, 34(2): 461-475 doi: 10.3934/dcds.2014.34.461 In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth$$ \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.
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