In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations ${\left(a+b{\int }_{\mathsf{\Omega }\times \mathsf{\Omega }}\frac{{|\xi \left(x\right)-\xi \left(y\right)|}^{{}^p}}{{|x-y|}^{N+ps(x,y)}}dxdy\right)}^{p-1}{(-\Delta )}_p^{s(·)}\xi +\lambda V\left(x\right){\left|\xi \right|}^{p-2}\xi =f(x,\xi ),x\in \mathsf{\Omega },\xi =0,x\in \partial \mathsf{\Omega },$ where $\mathsf{\Omega }$ is a bounded Lipschitz domain in ${\mathbb{R}}^N$, $1<p<+\infty$, $a,b>0$ are constants, $s(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ is a continuous and symmetric function with $N>s(x,y)p$ for all $(x,y)\in \mathsf{\Omega }\times \mathsf{\Omega }$, $\lambda >0$ is a parameter, ${(-\mathsf{\Delta })}_p^{s(·)}$ is a fractional p-Laplace operator with variable-order, $V\left(x\right):\mathsf{\Omega }\to {\mathbb{R}}^{+}$ is a potential function, and $f(x,\xi ):\mathsf{\Omega }\times {\mathbb{R}}^N\to \mathbb{R}$ is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

The paper investigates the following fractional Schrödinger equation: $$\begin{aligned} (-\Delta )^{s}u+V(x)u=K(x)f(u), \quad x\in \mathbb{R}^{N}, \end{aligned}$$ where \(0< s<1\), \(2s< N\), \((-\Delta )^{s}\) is the fractional Laplacian operator of order s. \(V(x)\), \(K(x)\) are nonnegative continuous functions and \(f(x)\) is a continuous function satisfying some conditions. The existence of infinitely many solutions for the above equation is presented by using a variant fountain theorem, which improves the related conclusions on this topic. The interesting result of this paper is the potential \(V(x)\) vanishing at infinity, i.e., \(\lim_{|x|\rightarrow +\infty }V(x)=0\).

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

In this paper, we investigate the fractional Schödinger equation involving a critical growth. By using the principle of concentration compactness and the variational method, we obtain some new existence results for the above equation, which improve the related results on this topic.

In this paper, an SEIR epidemic dynamical model incorporating a fear effect, a nonlinear incidence rate, and a time-delayed behavioral response was established. The basic reproduction number $ R_0 $ was derived using the next-generation matrix method. Theoretical analysis demonstrates that if $ R_{0} < 1 $, the disease-free equilibrium is globally asymptotically stable; if $ R_{0} > 1 $, there exists a unique endemic equilibrium, which is also globally asymptotically stable. By performing stability analysis, the critical time delay threshold $ \tau_0 $ for triggering a Hopf bifurcation was derived. Moreover theoretical analysis and numerical simulations consistently showed that when the time delay exceeds this threshold, the endemic equilibrium loses its stability and a stable limit cycle is generated, leading to periodic epidemic outbreaks. Furthermore, sensitivity analysis revealed an inverse relationship between fear coefficient $ \alpha $ and the critical time delay threshold $ \tau_0 $.