Related papers: Differential equations in automorphic forms

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

The eigenvalues and eigenfunctions of a linear {\alpha}^{2}-dynamo have been computed for different spatial distributions of an isotropic \alpha-effect. Oscillatory solutions are obtained when \alpha exhibits a sign change in the radial…

Near the singularity, gravity should be modified to an effective theory, in the same sense as with the Euler-Heisenberg electrodynamics. This effective gravity surmounts to higher derivative theory, and as is well known, a much more reacher…

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded…

We introduce new zeta functions related to an endomorphism $\phi$ of a discrete group $\Gamma$. They are of two types: counting numbers of fixed ($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of $\phi$ from an…

The full set of equations governing the structure and the evolution of self--gravitating spherically symmetric dissipative fluids with anisotropic stresses, is written down in terms of five scalar quantities obtained from the orthogonal…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^\alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where…

This is the third and last part of a series of 3 papers. Using the same method and the same coordinates as in parts 1 and 2, rotating dust solutions of Einstein's equations are investigated that possess 3-dimensional symmetry groups, under…

We find and study solutions to the Einstein equations in D dimensions coupled to a scalar field source with a Liouville potential under the assumption of D-2 planar symmetry. The general static or time-dependent solutions are found yielding…

We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several…

Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling the interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact and Hilbert--Schmidt operator…

Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$…

Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is…

The self-dual Einstein equation (SDE) is shown to be equivalent to the two dimensional chiral model, with gauge group chosen as the group of area preserving diffeomorphisms of a two dimensional surface. The approach given here leads to an…

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

Consider the equation $-\ve^2\Delta u_\ve+q(x)u_\ve=f(u_\ve)$ in $\R^3$, $|u(\infty)|<\infty$, $\ve=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_\ve$ exists and $\lim_{\ve\to 0} u_\ve=u(x)$, where…

It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental $s\mapsto 1-s$ invariance of $\zeta(s)$ by looking only at special values. In particular, via…

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-\Delta_p)^s u =|u|^{p^*_s-2}u+…