Related papers: Simultaneous equations and inequalities

In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

There are abundant results on Diophantine approximation over fields of positive characteristic (see the survey papers [13, 25]), but there is very little information about simultaneous approximation. In this paper, we develop a technique of…

In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u+ (-\Delta)^s_p u &= \lambda f(u) && \text{in } \Omega, u &= 0…

Here, we find all positive integer solutions of the Diophantine equation in the title, where $(\mathcal{U}_n)_{n\geqslant 0}$ is the generalized Lucas sequence $\mathcal{U}_0=0, \ \mathcal{U}_1=1$ and $\mathcal{U}_{n+1}=r \mathcal{U}_n +s…

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $\Omega$ in…

The paper is concerned with positive solutions to problems of the type \begin{equation*} -\Delta_{\mathbb{B}^N} u - \lambda u = a(x) |u|^{p-1}\;u \, + \, f \, \;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})},…

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a…

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…

We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such…

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas…

In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone…

We analyze the approximation by mixed finite element methods of solutions of equations of the form $-\mbox{div\,} (a\nabla u) = g$, where the coefficient $a=a(x)$ can degenerate going to cero or infinity. First, we extend the classic error…

In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for…

In this paper we present a new approach to prove effective results in Diophantine approximation. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with…

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n…

In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that…

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…