Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1<p<2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t\to \infty$.

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).

We study the leading order behaviour of positive solutions of the equation $-\Delta u+ϵu-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,x\in {ℝ}^N$, where $N\ge 3$, $q>p>2$ and when $ϵ>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*}=\frac{2N}{N-2}$. For $p<2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^{*}$ the solution asymptotically...

A new derivation of the classic asymptotic expansion of the $n$-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with ${li}^{-1}\left(n\right)$, after having retained the first $m$ terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n\&gt;s_3\left(n\right)$ where $s_3\left(n\right)$ is the sum of the first four terms of the asymptotic...

Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline{\delta }\left(S\right)$ denote the upper asymptotic density of $S$. We consider the problem of finding $$\mu \left(M\right):=\underset{S}{sup}\overline{\delta }\left(S\right),$$ where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b\notin M.$ In this paper we discuss the values and bounds of $\mu \left(M\right)$ where $M=\{a,b,a+nb\}$ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $gcd(a,b)=1.$

Let $c_q\left(n\right)$ be the Ramanujan sum, i.e. $c_q\left(n\right)={\sum }_{d\left|\right(q,n)}d\mu (q/d)$, where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for ${\sum }_{n\le y}{\left({\sum }_{q\le x}{c}_q\left(n\right)\right)}^k$ (k = 1,2) are obtained. As an analogous problem, we evaluate ${\sum }_{n\le y}{\left({\sum }_{n\le x}c{̂}_q\left(n\right)\right)}^k$ (k = 1,2), where $c{̂}_q\left(n\right):={\sum }_{d\left|\right(q,n)}d\left|\mu (q/d)\right|$.

Let $p,q\in ℝ$ with $p-q\ge 0$, $\sigma =\frac{1}{2}(p+q-1)$ and $s=\frac{1}{2}(1-p+q)$, and let $${𝒟}_m(x;p,q)={𝒟}_0(x;p,q)+\sum _{k=1}^m\frac{B_{2k}\left(s\right)}{2k{(x+\sigma )}^{2k}},$$ where $${𝒟}_0(x;p,q)=\frac{\psi (x+p)+\psi (x+q)}{2}-ln(x+\sigma ).$$ We establish the asymptotic expansion $${𝒟}_0(x;p,q)\sim -\sum _{n=1}^{\infty }\frac{B_{2n}\left(s\right)}{2n{(x+\sigma )}^{2n}}\text{as}x\to \infty ,$$ where $B_{2n}\left(s\right)$ stands for the Bernoulli polynomials. Further, we prove that the functions ${(-1)}^m{𝒟}_m(x;p,q)$ and ${(-1)}^{m+1}{𝒟}_m(x;p,q)$ are completely monotonic in $x$ on $(-\sigma ,\infty )$ for every $m\in {\mathbb{N}}_0$ if and only if $p-q\in [0,\frac{1}{2}]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.

We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure $m_k\left(P\right)$ of a polynomial $P,$ where $m_k\left(P\right)$ is the integral of $lo{g}^k\left|P\right|$ over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding $m_k\left(P\right)$, in particular $|m_k\left(P\right)|/k!\to 1/\pi$ as k → ∞.

We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $\epsilon cₜ=k_c\Delta c-f\left(c\right)c+g(a,c)$, x ∈ Ω, t > 0, for $\Omega \subset {ℝ}^N$, completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f\left(u\right)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that the omega limit set $\omega \left(u\right)$ of every bounded positive solution is determined by a stationary...

We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K/{\mathbb{Q}}_p}{GL}_2$ for $K$ a finite extension of ${\mathbb{Q}}_p$. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ which we obtain with a rate of convergence. ...