On the singularly perturbation fractional Kirchhoff equations: critical case

This paper deals with the following fractional Kirchhoff problem with critical exponent \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su=(1+\varepsilon K(x))u^{2^*_s-1}, \end{equation*} where $a,b>0$ are given constants, $\varepsilon$ is a small parameter, $2^*_s=\frac{2N}{N-2s}$ with $0<s<1$ and $N\geq4s$.

Non-degeneracy of positive solutions for fractional Kirchhoff problems: high dimensional cases

In this paper, we establish the nondegeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su+u=u^p, \end{equation*} where $a,b>0$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$ and $(-\Delta )^s$ is the fractional Laplacian. In particular, we prove that uniqueness breaks down for dimensions $N>4s$, i.

Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well

In this paper, we are concerned with the following eigenvalue problem \begin{equation*} (-\Delta)^su+\lambda g(x)u=\alpha u,\ \ u\in H^s(\mathbb{R}^N),\ N\geq3, \end{equation*} where $s\in(0,1),,\alpha,\lambda\in\mathbb{R}$ and \begin{equation*} g(x)\equiv0\ \text{on}\ \bar{\Omega},\ \ g(x)\in(0,1]\ \text{on}\ \mathbb{R}^N\backslash\bar{\Omega}\ \text{and}\ \lim_{|x|\rightarrow\infty}g(x)=1 \end{equation*} for some bounded open set $\Omega\subset\mathbb{R}^N$.

Multiplicity of solutions for a class of critical Schrödinger-Poisson system with two parameters

The following critical Schr"odinger-Poisson system is considered: \begin{equation*} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u, \end{equation*} \begin{equation*} -\Delta \phi=u^2, \end{equation*} where $\lambda, \mu$ are two positive parameters, $p\in(4,6)$ and $V$ satisfies some potential well conditions.

Existence and concentration of solution for Schrödinger-Poisson system with local potential

In this paper, we study the following nonlinear Schrödinger-Poisson type equation \begin{equation*} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u), \end{equation*} \begin{equation*} -\varepsilon^2\Delta \phi=K(x)u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $V: \mathbb{R}^3\rightarrow \mathbb{R}$ is a continuous potential and $K: \mathbb{R}^3\rightarrow \mathbb{R}$ is used to describe the electron charge.

Fujita exponent and nonexistence result for the Rockland heat equation

This article presents nonexistence results for semilinear parabolic equation with hypoelliptic operator. In particular, we show Fujita exponent for the Rockland heat equation on graded Lie group, which depends on the homogeneous dimension of group and the order of the Rockland operator.

Existence and concentration results for fractional Schrödinger-Poisson system via penalization method

In this paper, we are concerned with positive solutions for the fractional Schrödinger-Poisson system : \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u), \end{equation*} \begin{equation*} \varepsilon^{2t}(-\Delta)^t\phi=u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha=s,t\in(\frac{3}{4},1)$, $V\in C(\mathbb{R}^3,\mathbb{R})$ is the potential function and $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and subcritical.

Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

In this paper, we study the singularly perturbed fractional Choquard equation \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u=\varepsilon^{\mu-3}(\int_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u))\ \ \text{in}\ \mathbb{R}^3, \end{equation*} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes the fractional Laplacian of order $s\in(0,1)$, $0<\mu<3$, $2^*_{\mu,s}=\frac{6-\mu}{3-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.

Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials

In this paper, we study the following nonlinear elliptic systems: \begin{equation*} -\Delta u_1+V_1(x)u_1=\partial_{u_1}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*} \begin{equation*} -\Delta u_2+V_2(x)u_2=\partial_{u_2}F(x,u)\quad x\in \mathbb{R}^N, \end{equation*} where $u=(u_1,u_2):\mathbb{R}^N\rightarrow \mathbb{R}^2$, $F$ and $V_i$ are periodic in $x_1,\cdots,x_N$ and $0\notin \sigma(-\Delta+V_i)$ for $i=1,2$, where $\sigma(-\Delta +V_i)$ stands for the spectrum of the Schr"odinger operator $-\Delta +V_i$.

The fractional Schrödinger-Poisson systems with infinitely many solutions

In this paper, we study the existence of infinitely many large energy solutions for the supercubic fractional Schr"{o}dinger-Poisson systems. We consider different superlinear growth assumptions on the nonlinearity, starting from the well-know Ambrosetti-Rabinowitz type condition.