CPAA
Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system
Yanqin Fang Jihui Zhang
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: multiple solutions. Ljusternik-Schnirelmann theory Schrödinger-Maxwell
CPAA
Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation
Xudong Shang Jihui Zhang
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: Fractional Schrödinger equations Multiplicity of solutions Ekeland's variational principle
DCDS
Multi-peak positive solutions for a fractional nonlinear elliptic equation
Xudong Shang Jihui Zhang
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: Fractional elliptic equation Lyapunov-Schmidt reduction critical point. multi-peak solutions
CPAA
Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$
Yanqin Fang Jihui Zhang
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: monotonicity. moving planes in integral forms Nonexistence
CPAA
Symmetry and nonexistence of positive solutions for fractional systems
Pei Ma Yan Li Jihui Zhang
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
$1<p<\frac{n+α+a}{n-α}$
and
$1<q<\frac{n+α+b}{n-α}$
.
keywords: The method of moving planes fractional Laplacian nonlinear elliptic system radial symmetry nonexistence of positive solutions
CPAA
Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
Xudong Shang Jihui Zhang Yang Yang
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.
keywords: Positive solutions fractional Laplacian critical exponent.

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