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
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
$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$
is a positive parameter,
is the fractional Laplacian,
$s ∈ (0,1)$
$N> 2s$
. Suppose that the potential
$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$
$\text{inf}_{\mathbb{R}^{N}} V(x)>0$
, and there exist
$x^{j} ∈ \mathbb{R}^{N}$
such that for each
$j = 1,···,k$
are strict global minimum. When
is subcritical, we prove that the problem has at least
positive solutions for
small. Moreover, we establish the concentration property of the solutions as
tends to zero.
keywords: Fractional Schrödinger equations Multiplicity of solutions Ekeland's variational principle
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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