By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: $M\left({\left[u\right]}_{s,p}^p+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx\right)[{(-\Delta )}_p^{s}u+$${V\left(x\right)\left|u\right|}^{p-2}u]={\lambda \left|u\right|}^{q-2}u\ln \left|u\right|,x\in {\mathbb{R}}^N,\left(\mathrm{P}\right)$. In these equations, $\lambda \in \mathbb{R}\setminus \left\{0\right\},$$p\in (1,+\infty )$, $s\in (0,1),sp<N,$$p_s^{*}={\displaystyle \frac{Np}{N-sp}}$, $M\left(\tau \right)=a+b{\tau }^{\theta -1}$, $a,b\in {\mathbb{R}}^{+},$$1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$, $V\left(x\right)\in C({\mathbb{R}}^N,\mathbb{R})$ is a potential function and ${(-\Delta )}_p^s$ is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of $\lambda$. More precisely, (i) Equation (P) has one ground state solution for $\lambda >0$ and $p\theta <q<p_s^{*}$, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for $\lambda <0$ and $p<q<p_s^{*}$, with positive and negative corresponding energy values, respectively.

By using 35 cities data directly under the jurisdiction of Beijing-Tianjin-Hebei region, this paper improved Bayesian spatial quantile regression method to explore the relationship between population aggregation and eco-efficiency and its mutual influence under the background of Beijing, Tianjin and Hebei urban agglomeration construction theme. The study found that population aggregation has a significant negative impact on the urban eco-efficiency. In addition, the number of primary schools, the scale of urban economic development and investment in real estate development also have a significant negative impact on urban eco-efficiency. While the number of secondary schools and the number of beds in medical and health institutions has a significant positive impact on urban eco-efficiency.

We consider the following nonlinear problem u″x+λux+fx,u=0, x∈S1, (P) where S1 is a unit circle, λ∈ℝ is a parameter, and fx,u is a nonlinearity. By using the variational method, we obtain the existence of a positive solution to (P) for all λ∈ℝ. This phenomenon is different from the corresponding boundary value problem.