Related papers: Allen-Cahn Solutions with Triple Junction Structur…
In this note, we deal with a problem of the type $$\cases {-h\left ( \int_{\Omega}|\nabla u(x)|^2dx\right ) \Delta u=f(u) & in $\Omega$\cr & \cr u_{|\partial\Omega}=0\ .\cr}$$ As an application of a new general multiplicity result, we…
The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of It\^o type. The problem is set in a bounded…
In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schr\"odinger system \begin{align*} \begin{gathered} -\Delta u + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q…
In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -\Delta u=|u|^{2^*-2}u+\lambda u&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball…
In this work, we prove the existence of least energy nodal solution for a class of elliptic problem in both cases, bounded and unbounded domain, when the nonlinearity has exponential critical growth in $\mathbb{R}^2$. Moreover, we also…
This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed.…
We consider the 3D smectic energy $$\mathcal{E}_{\epsilon }\left( u\right) =\frac{1}{2}\int_{\Omega }\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+( u_y)^{2}}{2}\right) ^{2}+\varepsilon \left( u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$…
In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-\Delta)^s u &= \displaystyle-\lambda|u|^{q-2}u +…
We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+\kappa u-\mu u^2$ $ 0 =\Delta v-v+u$ $\partial_\nu v|_{\partial\Omega} = \partial_\nu u|_{\partial\Omega}=0,$ $…
We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $\Delta_{p}u =…
We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be…
We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see…
We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and…
We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This…
For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions,…
In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic space $\H^m$ was reduced to the problem of constructing a minimal-energy blowup solution…
We study a linearly coupled Schr\"{o}dinger system in $\R^N(N\leq3).$ Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and the parameters in the…
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space…
The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \left\{\begin{array}{rcll} (-\Delta)^s u-\lambda \dfrac{u}{|x|^{2s}}&=&f(x,u)…
In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by…