Related papers: Bifurcation diagrams of one-dimensional Kirchhoff …

In this paper we are concerned with the following Kirchhoff type problem involving the 1-Laplace operator : \begin{equation*} \left\{\begin{array}{llc} u_{t}-m\left(\int_{\Omega}|Du|\right)\Delta_{1} u=0 & \text{in}\ & \Omega\times…

Using Leray-Schauder degree theory we study the existence of at least one solution for the boundary value problem of the type (\varphi(u' ))' = f(t,u,u'), u'(0)=u(0), u'(T)= bu'(0), where \varphi is a homeomorphism such that \varphi(0)=0, f…

In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem…

We study a class of boundary value problems with $\varphi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\varphi(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u')\right)'=\lambda f(u)\; \text{ on }(-L,…

Asymptotics of the eigenvalues can always be derived for self-adjoint boundary value problems. However, they can also be derived for boundary value problems that fail to be self-adjoint provided that they are Birkhoff regular. A regular…

In this paper we consider the following Sturm-Liouville equation \[ \left\{ \begin{aligned} -(x^{2\alpha}u'(x))'+u(x)&=f(x) && \text{in } (0,1],\\ u(1)&=0 \end{aligned} \right. \] where $\alpha<1$ is a nonzero real number and $f$ belongs to…

We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right) u=f(x,u) &…

In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})\Delta u = f(u) & \mbox{in $\Omega$,} u=0 & \mbox{on $\partial\Omega$,} \end{array}\right. \end{equation*} in a bounded domain,…

We discuss, via a version of the Birkhoff-Kellogg theorem, the existence of positive and negative eigenvalues of Hammerstein integral equations with sign-changing nonlinearities and functional terms. The corresponding eigenfunctions have a…

We study the following elliptic problem $-A(u) = \lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = \Delta u (x) \chi_{D_1} (x)+ \Delta_p u(x) \chi_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the…

In this paper, we discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation $$h(x,d_x u)+\lambda(x)u=c,\quad x\in M,$$ where $M$ is a closed manifold and $\lambda:M\rightarrow\mathbb{R}$ changes…

We perform a bifurcation analysis of the steady states of Rayleigh--B\'enard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an…

We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We…

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u &=|x|^\alpha |u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on…

Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…

We consider existence and uniqueness of homogeneous solutions $ u > 0 $ to certain PDE of $p$-Laplace type, $ p $ fixed, $ n - 1 <p< \infty, n \geq 2, $ when $ u $ is a solution in $K(\alpha)\subset\mathbb{R}^n$ where \[ K (\alpha) := \{ x…

In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$…

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf…

In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 & on\quad \partial\Omega\\ \end{array} \right.…

The study on the partial differential equations (systems) in the graph setting is a hot topic in recent years because of their applications to image processing and data clustering. Our motivation is to develop some existence results for…