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### Open Access Journals

DCDS

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 $\beta > 0$.

DCDS

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.

DCDS

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].
DCDS

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.

DCDS

Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

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.

DCDS

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 $(\star)$.

DCDS

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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