Related papers: Hyperexponential solutions of elliptic difference …

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such…

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D. Browning and…

In this paper, we consider the following nonlinear elliptic equation with gradient term: \[ \left\{ \begin{gathered} - \Delta u - \frac{1}{2}(x \cdot \nabla u) + (\lambda a(x)+b(x))u = \beta u^q +u^{2^*-1}, \hfill 0<u \in…

We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron…

Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group…

We compute the rational points on certain members of the following family of hyperelliptic curves \[C_a \colon y^2 = x^8 + (4-4a^4) x^6 + (8a^4 + 6)x^4 + (4-4a^4)x^2 + 1\] via the method first developed by Dem'yanenko \cite{dem1966rational}…

L-function and rational points on an elliptic curve via the classical number theory.

We show how the basic idea of parabolic Jacobi relaxation can be modified to obtain a new class of hyperbolic relaxation schemes that are suitable for the solution of elliptic equations. Some of the analytic and numerical properties of…

In this paper, we study the existence and regularity of solutions for a class of nonlinear singular elliptic equations involving unbounded coefficients and a singular right-hand side. Specifically, we are interested to problem whose…

We deal with the following semilinear equation in exterior domains \[-\Delta u + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. Assuming that the weight $a$ is positive and…

In this paper we consider the curves $H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1}$ over $\mathbb F_p$ and and find an exact formula for the number of $\mathbb F_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give…

Consider a pair of ordinary elliptic curves $E$ and $E'$ defined over the same finite field $\mathbb{F}_q$. Suppose they have the same number of $\mathbb{F}_q$-rational points, i.e. $|E(\mathbb{F}_q)|=|E'(\mathbb{F}_q)|$. In this paper we…

A new variational approach to solve the problem of estimating the (possibly discontinuous) coefficient functions $p$, $q$ and $f$ in elliptic equations of the form $-\nabla \cdot (p(x)\nabla u) + \lambda q(x) u = f$, $x \in \Omega \subset…

In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these…

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -\Delta u =\lambda f(v) \quad \mbox{ in } \Omega -\Delta v =\gamma f(u) \quad \mbox{ in } \Omega, u=v =0, \quad \mbox{ on } \pOm \end{array}\right.…

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$.…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…