Related papers: On Two Nonlinear Difference Equations
Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for two-parametric family of partially homogeneous $(p,q)$-Laplace equations $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u+\beta |u|^{q-2}u$ where $p \neq…
Asymptotic properties of solutions of difference equation of the form \[ \Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n \] are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or…
In this paper, we study nonlinear differential equations of Tumura-Clunie type, $ f^n + P(z, f) = h, $ where \( n \geq 2 \) is an integer, \( P(z, f) \) is a differential polynomial in \( f \) of degree \( \gamma_P \leq n - 1 \) with small…
In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…
The meromorphic solutions $f$ with $\rho_2(f)<1$ of the non-linear difference equation \begin{align*} f^n(z)+P_d(z,f)=p_1e^{{\lambda_1}z}+p_2e^{{\lambda_2}z}+p_3e^{{\lambda_3}z}, \end{align*} are characterized in terms of exponential…
Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and…
In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \geq 1$, $ A,…
The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u - u^p,$ $(\alpha\in(0,2], p>1)$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is…
In the present work we shall consider the existence and multiplicity of solutions for nonlocal elliptic singular problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following…
We consider the existence of multiple positive solutions to the nonlinear Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…
This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) =…
We investigate a nonlinear nonlocal eigenvalue problem involving the sum of fractional $(p,q)$-Laplace operators $(-\Delta)_p^{s_1}+(-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$; $p,q\in(1,\infty)$ and subject to Dirichlet boundary conditions…
We study properties of nonnegative functions satisfying (E)$\;-\Delta u+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1<q<p$. We concentrate our analysis on the solutions of (E) with an isolated singularity, or…
We consider positive solutions of the problem \begin{equation} \left\{\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=0\qquad \mbox{in}\;\;\mathbb{R}_+^n,\\ \frac{\partial u}{\partial \nu^a}=u^{q} \qquad \mbox{on}\;\;\partial…
Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbb{N}$. First, we study singularities of…
We improve earlier work on the title equation (where $p$ and $q$ are primes and $c$ is a positive integer) by allowing $x$ and $y$ to be zero as well as positive. Earlier work on the title equation showed that, with listed exceptions, there…
In this paper, we study the nonlinear Choquard equation \begin{eqnarray*} \Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u \end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth with the…
For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their…