# American Institute of Mathematical Sciences

## Journals

CPAA
Communications on Pure & Applied Analysis 2011, 10(4): 1267-1279 doi: 10.3934/cpaa.2011.10.1267
In this paper, we study the following system

$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

keywords:
CPAA
Communications on Pure & Applied Analysis 2018, 17(6): 2239-2259 doi: 10.3934/cpaa.2018107
In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation
 \left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.
where
 $\varepsilon$
is a positive parameter,
 $(-Δ)^{s}$
is the fractional Laplacian,
 $s ∈ (0,1)$
and
 $N> 2s$
. Suppose that the potential
 $V(x) ∈\mathcal{C}(\mathbb{R}^{N})$
satisfies
 $\text{inf}_{\mathbb{R}^{N}} V(x)>0$
, and there exist
 $k$
points
 $x^{j} ∈ \mathbb{R}^{N}$
such that for each
 $j = 1,···,k$
,
 $V(x^{j})$
are strict global minimum. When
 $f$
is subcritical, we prove that the problem has at least
 $k$
positive solutions for
 $\varepsilon>0$
small. Moreover, we establish the concentration property of the solutions as
 $\varepsilon$
tends to zero.
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 3183-3201 doi: 10.3934/dcds.2015.35.3183
In this paper we study the existence of positive multi-peak solutions to the semilinear equation \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N}) \end{eqnarray*} where $(-\Delta)^{s}$ stands for the fractional Laplacian, $s\in (0,1)$, $\varepsilon$ is a positive small parameter, $2 < p < \frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function. For any positive integer $k$, we prove the existence of a positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].
keywords:
CPAA
Communications on Pure & Applied Analysis 2013, 12(2): 663-678 doi: 10.3934/cpaa.2013.12.663
Let $n, m$ be a positive integer and let $R_+^n$ be the $n$-dimensional upper half Euclidean space. In this paper, we study the following integral equation \begin{eqnarray} u(x)=\int_{R_+^n}G(x,y)u^pdy, \end{eqnarray} where \begin{eqnarray*} G(x,y)=\frac{C_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_n y_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{n/2}}dz, \end{eqnarray*} $C_{n}$ is a positive constant, $0 <2m 1$. Using the method of moving planes in integral forms, we show that equation (1) has no positive solution.
keywords:
CPAA
Communications on Pure & Applied Analysis 2018, 17(3): 1053-1070 doi: 10.3934/cpaa.2018051
We consider the following fractional Hénonsystem
 $\left\{ \begin{array}{*{35}{l}} {}&{{(-\vartriangle )}^{\alpha /2}}u = |x{{|}^{a}}{{v}^{p}},~~&x\in {{R}^{n}}, \\ {}&{{(-\vartriangle )}^{\alpha /2}}v = |x{{|}^{b}}{{u}^{q}},~~&x\in {{R}^{n}}, \\ {}&u\ge 0,v\ge 0,&{} \\\end{array} \right.$
for
 $0<α<2$
and
 $a, b$
 $≥0$
,
 $n≥2$
. Under rather weaker assumptions, by using a direct method of moving planes, we prove the nonexistence and symmetry of positive solutions in the subcritical case where
 $1 and $1
.
keywords:
CPAA
Communications on Pure & Applied Analysis 2014, 13(2): 567-584 doi: 10.3934/cpaa.2014.13.567
In this paper, we study the following problem \begin{eqnarray} (-\Delta)^{\frac{\alpha}{2}}u = K(x)|u|^{2_{\alpha}^{*}-2}u + f(x) \quad in \ \Omega,\\ u=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega\subset R^N$ is a smooth bounded domain, $0< \alpha < 2$, $N>\alpha$, $2_{\alpha}^{*}= \frac{2N}{N-\alpha}$, $f\in H^{-\frac{\alpha}{2}}(\Omega)$ and $K(x)\in L^\infty(\Omega)$. Under appropriate assumptions on $K$ and $f$, we prove that this problem has at least two positive solutions. When $\alpha = 1$, we also establish a nonexistence result for a positive solution in a class of linear positive-type domains are more general than star-shaped ones.
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