In this work we consider the fractional Kirchhoff equations with singular nonlinearity, $$\displaylines{ M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx dy\Big) (-\Delta)^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\cr \int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}dx v \mu b(x)|v|^{q-2}v +\fra...

In this paper, we derive a sharp observability inequality for Kirchhoff plate equations with lower order terms in an unbounded domain Ω of lR. More precisely, when the observation is assumed to be located in a subdomain ω such that Ω \ ω is bounded and the observation time T > 0 is sufficiently large, we establish an observability estimate with an explicit observability constant for Kirchhoff p...

In this paper, existence of solutions is established for critical exponential Kirchhoff systems on the Heisenberg group by using variational method. The novelty our paper that not only nonlinear term has growth, but also function covers degenerate case. Moreover, result new even Euclidean

In this article, we are interested in the existence of positive solutions for the following Kirchhoff type system  −M1 (∫ Ω |x| −ap|∇u|p dx ) div(|x|−ap|∇u|p−2∇u) = λ|x|−(a+1)p+c1 (f(v)− 1 uα ) in Ω, −M2 (∫ Ω |x| −bq |∇v|q dx ) div(|x|−bq |∇v|q−2∇v) = λ|x|−(b+1)q+c2 (g(u)− 1 vβ ) in Ω, u = v = 0 on ∂Ω, (1) where Ω is a bounded smooth domain of R with 0 ∈ Ω, 1 < p, q < N , 0 ≤ a, b < N−p p , c...

The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the n-vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for n ≥ 5. We also ...