The aim of this paper is to study the existence of ground states for a class of fractional Kirchhoff type equations with critical or supercritical nonlinearity

$$(a+b{\int }_{{\mathbb{R}}^3}|(-△{)}^{\frac{s}{2}}u{|}^{2}dx)(-△{)}^{s}u=\lambda u+|u{|}^{q-2}u+\mu |u{|}^{p-2}u,x\in {\mathbb{R}}^3,$$

with prescribed $L^2$-norm mass

$${\int }_{{\mathbb{R}}^3}{u}^{2}dx=c^2$$

where $s\in (\frac{3}{4},1),a,b,c>0,\frac{6+8s}{3}<q<2_s^{\ast },p\ge 2_s^{\ast }(2_s^{\ast }=\frac{6}{3-2s}),\mu >0$ and $\lambda \in \mathbb{R}$ as a Langrange multiplier. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of normalized solutions for the above equation when the parameter $\mu$ is sufficiently small.

In this paper, we investigate the existence of sign-changing and signed solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical or supercritical nonlinearity. By combining an appropriate truncation argument with a constrained minimization method and the Moser iteration method, we obtain a sign-changing solution and a signed solution for them under some suitable assumptions. As a particular case, we drive an existence theorem of sign-changing and signed solutions for the fractional Laplacian equations with critical or supercritical growth.

In this paper, the Cauchy problem for a class of reaction diffusion equations are considered with nonlocal interactions in periodic media. First, we demonstrate the existence and uniqueness of solutions that are both positive and bounded for the stationary equation. Second, we derive results concerning the existence and uniqueness of solutions for the Cauchy problem by using the semigroup theory. Finally, we analyze the behavior of the solutions to the Cauchy problem for large times by using the comparison principle.