Related papers: Existence of a solution to a vector-valued Allen-C…

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha…

We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are…

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in…

In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary…

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem…

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in…

We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and…

We consider the equation $\e^{2}\Delta u=(u-a(x))(u^2-1)$ in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to $\pa\Omega$, and $a$ a…

We outline a regular way for solving Maxwell's equations. We take, as the starting point, the notion of vector potentials. The rationale for introducing this notion in electrodynamics is that the set of Maxwell's equations is seemingly…

In this paper we find solutions $u_\epsilon$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $\epsilon \to 0$ develops as interface a given triod evolving under curve shortening flow.

We construct weak solutions to the 3D hypoviscous incompressible elastodynamics with finite kinetic energy which was unknown in literatures. Our result holds for fractional hypoviscosity $(-\Delta)^\theta$, where $0\leq\theta<1$. The proof…

In this paper, by using variational methods we study the existence of positive solutions for the following Kirchhoff type problem: $$ \left\{ \begin{array}{ll} -\left(a+b\mathlarger{\int}_{\Omega}|\nabla u|^{2}dx\right)\Delta u+V(x)u=u^{5},…

We prove quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle…

We prove that sufficiently regular solutions to the wave equation $\Box_g\phi=0$ on the exterior of the Schwarzschild black hole obey the estimates $|\phi|\leq C_\delta v_+^{-{3/2}+\delta}$ and $|\partial_t\phi|\leq C_{\delta}…

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$…

Energy positivity is established for a class of solutions to Einstein-aether theory and the IR limit of Ho\v{r}ava gravity within a certain range of coupling parameters. The class consists of solutions where the aether 4-vector is…

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…

We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions…

We show that weak solutions of the relativistic Vlasov-Maxwell system preserve the total energy provided that the electromagnetic field is locally of bounded variation and, for any $\lambda$> 0, the one-particle distribution function has a…

In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} -\Delta - \nabla \cdot (a-I) \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - \Delta - (a-I)…