Related papers: Bifurcation diagrams of one-dimensional Kirchhoff …

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with convolutional Kirchhoff functions. We establish the exact solutions $u_\lambda$ and bifurcation curves $\lambda(\alpha)$, where $\alpha:= \Vert…

We study the one-dimensional nonlocal elliptic equation \begin{eqnarray*} -\left(\int_0^1 \vert u(x)\vert^p dx + b\right)^q u''(x) &=& \lambda u(x)^p, \quad x \in I:= (0,1), \ u(x) > 0, \ x\in I, \\ u(0) &=& u(1) = 0, \end{eqnarray*} where…

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda…

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $u_\lambda(x)$ as $\lambda \to \infty$. Here, $\lambda > 0$ is the…

This paper is concerned with the existence and uniqueness of positive solution for the fourth order Kirchhoff type problem $$\left\{\begin{array}{ll} u''''(x)-(a+b\int_0^1(u'(x))^2dx)u''(x)=\lambda f(u(x)),\ \ \ \ x\in(0,1),\\…

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with oscillatory nonlinear term. We establish the precise asymptotic formulas for the bifurcation curves $\lambda(\alpha)$ as $\alpha \to \infty$ and $\alpha \to 0$,…

We study the one-dimensional nonlocal Kirchhoff type bifurcation problem related to logistic equation of population dynamics. We establish the precise asymptotic formulas for bifurcation curve $\lambda = \lambda(\alpha)$ as $\alpha \to…

We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $\lambda$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical…

In this paper, we study one-dimensional boundary blow up problems with Kirchhoff type nonlocal terms on an interval. We perform a bifurcation analysis on the problems and obtain the precise number of solutions according to the value of the…

The one-dimensional nonlocal Kirchhoff type bifurcation problems which are derived from logistic equation of population dynamics are studied. We obtain the precise asymptotic shapes of $L^2$ bifurcation curves $\lambda = \lambda(\alpha)$ as…

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\…

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array}…

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem $-\left(|\phi_x|^{p-2}\phi_x\right)_x=\lambda \left(|\phi|^{q-2}\phi-f(\phi)\right)$ such that $\phi(0)=\phi(1)=0$, where $p,q>1$, $\lambda$ is a…

We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…

We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly…

In this paper, we consider the following eigenvalue problem {{l} (|u"|^{p-2}u")"=\lambda m(x)|u|^{p-2}u, x\in (0,1), u(0)=u(1)=u"(0)=u"(1)=0, where $1<p<+\infty$, $\lambda$ is a real parameter and $m$ is sign-changing weight. We prove there…

We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{\lambda,\mu}$ is a step-wise…

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary…

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $\lambda \in…

In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\%…