Related papers: Saddle solutions for Allen-Cahn type equations inv…

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type.…

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla…

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation $(-\Delta)^{1/2}u=f(u)$ in all the space $\re^{2m}$, where $f$ is of bistable type. These solutions are odd with respect to the…

We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We…

In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where…

We consider a class of semilinear elliptic system of the form $-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in\R^{2}$ where $W:\R^{2}\to\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the…

An entire solution of the Allen-Cahn equation $\Delta u=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$…

This is the first of two papers concerning saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator and $f$ is of Allen-Cahn type. Saddle-shaped…

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in \mathbb{R}^N \end{align*} where $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential…

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system \[…

In the spirit of Berestycki and Lions, we prove the existence of saddle type nodal solutions for the Choquard equation \[ -\Delta u + u= \big(I_\alpha \ast F(u)\big)F'(u)\qquad \text{ in }\;\mathbb{R}^N \] where $N\geq 2$ and $I_\alpha$ is…

In this paper we prove the uniqueness of the saddle-shaped solution to the semilinear nonlocal elliptic equation $(-\Delta)^\gamma u = f(u)$ in $\mathbb R^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen-Cahn type. Moreover, we prove…

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} \Delta u-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0…

In this manuscript, we derive Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations \[ \partial_t u - F(x, t,D^2u) = f (x, t) \quad \text{in} \quad \mathrm{Q}_1 = B_1 \times (-1, 0], \]…

In this paper we construct entire solutions to the Cahn-Hilliard equation $-\Delta(-\Delta u+W^{'}(u))+W^{"}(u)(-\Delta u+W^{'}(u))=0$ in the Euclidean plane, where $W(u)$ is the standard double-well potential $\frac{1}{4} (1-u^2)^2$. Such…

In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation $\ep^2\div\left(a(x)\nabla u\right)-a(x)F'(u)=0$ in $\R^2$, where $\ep\to 0$. The nodal set of these solutions is…

This paper presents a conditional convergence result of solutions to the Allen--Cahn equation with arbitrary potentials to a De Giorgi type $ \mathrm{BV} $-solution to multiphase mean curvature flow. Moreover we show that De Giorgi type…

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: \[ -\Delta_{\mathbb H} u+F'(u)=0; \] here $F$ is a nonnegative double-well potential with nondegenerate minima. We…

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori…

The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean…