Abstract In this paper we study the existence of positive solution to Kirchhoff elliptic problem $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \lambda f(u) &amp;{} \text{ in } \; \Omega ,\\ 0 on \partial \end{array}\right. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML...

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...

In this work, we investigate the following Kirchhoff-type equation with variable exponent nonlinearities u_{tt}-M(‖∇u‖²)△u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u. We proved the blow up of solutions in finite time by using modified energy functional method.