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 $N\geq4s$.
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.
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 $\Omega\subset\mathbb{R}^N$.
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.
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.
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.
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.
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.
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 $-\Delta +V_i$.
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.