in this paper‎, ‎we consider the following kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{r}^{3}}|nabla u|^{2}right)delta u+v(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{r}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{r}^{3},$ ‎$uin h^{1}(mathbb{r}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎the aim of this paper is to study the existence of positive ...

In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive ...

Abstract We study the one-dimensional Kirchhoff-type equation − ( b + a ‖ u accent="false">′ 2 ) accent="true">″ x = <...

In this paper, we study the initial boundary value problem of important hyperbolic Kirchhoff equation $$\begin{aligned} u_{tt}-\left( a \int _\text{\O}mega |\nabla u|^2 \mathrm {d}x +b\right) \Delta u = \lambda u+ |u|^{p-1}u , \end{aligned}$$ where a, $$b>0$$ $$p>1$$ $$\lambda \in {\mathbb {R}}$$ and energy is arbitrarily large. We prove several new theorems on dynamics such as boundedness or f...

and many authors have studied the existence and uniqueness of global solution, the blowup of the solution (see [–] and the references therein). WhenM is not a constant function, equation (.)without the damping and source terms is often called a Kirchhoff-type wave equation; it has first been introduced by Kirchhoff [] in order to describe the nonlinear vibrations of an elastic string. When...