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
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
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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