It is well known that Kirchhoff equations admit infinitely many simple modes, i. e. time periodic solutions with only one Fourier component in the space variables. We prove that, for some choices of the nonlinearity, these simple modes are unstable provided that their energy is large enough. This result, stated in an abstract Hilbert space setting, and proved by reducing to a system of two seco...

We study solution methods for boundary value problems associated with the static Kirchhoff rod equations. Using the well known Kirchhoff kinetic analogy between the equations describing the spinning top in a gravity field and spatial rods, the static Kirchhoff rod equations can be fully integrated. We first give an explicit form of a general solution of the static Kirchhoff equations in paramet...

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \begin{cases} \displaystyle - \Big ( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b ) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\[2mm] u\in H^{1}(\mathbb{R}^{N}), \end{cases} \end{equation*} where $N\geq 1,a,b > 0, 2 < p \min \left\{ 4,2^{\ast }\right\}$($2^{\ast }=\infty $ for $N=1,2$ and $2^...

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions with help variational methods, reach positive energy solution negative in an appropriate space functions. The main difficulties innovations are Choquard nonlinearities Kirchhoff functions presence double Laplace ...