相关论文: Multiple Delaunay ends solutions of the Cahn-Hilli…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

In this paper we discuss nondegeneracy and stability properties of some special minimal hypersurfaces which are asymptotic to a given Lawson cone $C_{m,n}$, for $m,\,n\ge 2$. Then we use such hypersurfaces to construct solutions to the…

Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma…

It has been showed by Byde that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The…

We consider a one dimensional periodic forward-backward parabolic equation, regularized by a non-linear fourth order term of order $\epsilon^2\ll 1$. This equation is known in the literature as Cahn-Hilliard equation with degenerate…

In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as $\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation \[ H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad…

This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers…

In this paper we study the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}}…

In this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…

In this article, we study the following Hamiltonian system: \begin{equation*} \begin{cases} \begin{aligned} &-\varepsilon^{2}\Delta_{g} u +u = |v|^{q-1}v, &-\varepsilon^{2}\Delta_{g} v +v = |u|^{p-1}u && \text{ in } \mathcal{M}, & \quad u,v…

Let $\Omega$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partial\Omega$. We consider the equation $ \Delta u + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,\Omega $, under zero Dirichlet boundary condition,…

We use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport for- mulae and…

In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t - u_{xxt} =…

In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -\Delta u = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad \mathbb{R}^n, \quad u \in D^{1,2}(\mathbb{R}^n), \end{equation*} where $K(x)$…

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence…

We study solutions of Hamilton--Jacobi equations of the form $$\lambda \alpha(x) u_\lambda(x) + H(x, D_x u_\lambda) = c,$$ where $\alpha$ is a nonnegative function, $\lambda$ a positive constant, $c$ a constant and $H $ a convex coercive…

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our…

The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

We consider the following prescribed scalar curvature equations in ${\mathbb{R}}^N$ $$ - \Delta u =K(|y|)u^{2^*-1},\quad u>0 \quad \mbox{in} \quad {\mathbb{R}}^N, \quad u \in D^{1, 2}({\mathbb{R}}^N), $$ where $K(r)$ is a positive function,…