Jacobsthal sums and permutations of biquadratic residues

Let $p\equiv 1(\mathrm{mod}4)$ be a prime. In this paper, with the help of Jacobsthal sums over finite fields, we study some permutation problems involving biquadratic residues modulo p.     

Geometrically distinct solutions for Klein–Gordon–Maxwell systems with super-linear nonlinearities

This paper is concerned with the following Klein–Gordon–Maxwell system $\left\{\begin{array}{c}-△u+V\left(x\right)u-(2\omega +\varphi )\varphi u=f(x,u),x\in {\mathbb{R}}^3,\hfill \\ △\varphi =(\omega +\varphi )u^2,x\in {\mathbb{R}}^3,\hfill \end{array}\right.$where $\omega >0$ is a constant, $V$ and $f$ are periodic with respect to $x$. By combining deformation type arguments, Lusternik–Schnirelmann theory and some new tricks, we prove that the above system admits infinitely many geometrically distinct solutions under weaker superlinear conditions instead of the common super-cubic conditions on $f$. Our result seems new and extends the previous results in the literature.     

Non-Wieferich primes under the abc conjecture

Assuming the abc conjecture, Silverman proved that, for any given positive integer $a?2$, there are $?\log ?x$ primes $p?x$ such that $a^{p?1}?1(\mathrm{mod}{p}^2)$. In this paper, we show that, for any given integers $a?2$ and $k?2$, there still are $?\log ?x$ primes $p?x$ satisfying $a^{p?1}?1(\mathrm{mod}{p}^2)$ and $p\equiv 1(\mathrm{mod}k)$, under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.     

The spectrum of resolvable holey Mendelsohn triple systems and holey Mendelsohn frames with block size three

A holey Mendelsohn triple system (HMTS) is a decomposition of a complete multipartite directed graph into directed cycles of length 3. If the directed cycles of length 3 can be partitioned into parallel classes, then the HMTS is called an RHMTS. Bennett, Wei and Zhu [J. Combin. Des., 1997] showed that an RHMTS of type $g^n$ exists when $gn\equiv 0(\mathrm{mod}3)$ and $(g,n)\ne (1,6)$ with some possible exceptions. In this paper, motivated by the application in constructing RHMTSs, we investigate the constructions of holey Mendelsohn frames. We prove that a 3-MHF of type $(n,h^t)$ exists if and only if $n\ge 3$, $t\ge 4$ and $nh(t-1)\equiv 0(\mathrm{mod}3)$, and then determine that the necessary condition for the existence of an RHMTS of type $g^n$, namely, $gn\equiv 0(\mathrm{mod}3)$ is also sufficient except for $(g,n)=(1,6)$. New recursive constructions on incomplete RHMTSs via MHFs are introduced to settle this problem completely.     

The spectrum for self-converse large set of pure Mendelsohn triple systems

An $LPMTS(v,\lambda )$ is a collection of $\frac{v-2}{\lambda }$ disjoint pure Mendelsohn triple system $PMTS(v,\lambda )s$ on the same set of v elements. An $LPMT{S}^{⁎}(v)$ is a special $LPMTS(v,1)$ which contains exactly $\frac{v-2}{2}$ converse pairs of $PMTS(v)s$. In this paper, we mainly discuss the existence of an $LPMT{S}^{⁎}(v)$ for $v\equiv 6,10\mathrm{mod 12}$ and get the following conclusions: (1) there exists an $LPMT{S}^{⁎}(v)$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 4$ and $v\ne 6$. (2) There exists an $LPMTS(v,\lambda )$ with index $\lambda \equiv 2,4\mathrm{mod 6}$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 2\lambda +2,v\equiv 2\mathrm{mod}\mathit{\lambda }$.     

Ground state solutions for Klein–Gordon–Maxwell system with steep potential well

In this paper, we study the following Klein–Gordon–Maxwell system $\left\{\begin{array}{c}-\Delta u+(\lambda a\left(x\right)+1)u-(2\omega +\varphi )\varphi u=f(x,u),\mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\mathrm{in}{\mathbb{R}}^3.\hfill \end{array}\right.$Using variational methods, we obtain the existence of ground state solutions under some appropriate assumptions on $a$ and $f$.     

Self-converse large sets of pure directed triple systems

An $LPDTS(v,\lambda )$ is a collection of $\frac{3(v?2)}{\lambda }$ pairwise disjoint $PDTS(v,\lambda )s$ on the same set of v elements. An $LPDT{S}^{?}(v)$ is a special $LPDTS(v,1)$ which contains exactly $\frac{v?2}{2}$ converse hexads of $PDTS(v)s$. In this paper, we mainly discuss the existence of an $LPDT{S}^{?}(v)$ and get the following conclusions: (1) there exists an $LPDT{S}^{?}(v)$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 4$ except possibly $v=30$. (2) There exists an $LPDTS(v,\lambda )$ with index $\lambda \equiv 2,4\mathrm{mod 6}$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 2\lambda +2,v\equiv 2\mathrm{mod}\mathit{\lambda }$ except possibly $v=30$.     

Symmetric results of a Hénon type elliptic equation

In this paper, we study the following weighted elliptic system $\left\{\begin{array}{cc}\hfill & -\Delta u+u={\lambda }_1{\left|x\right|}^{\alpha }{u}^3+\mu {\left|x\right|}^{\beta }u{v}^2,x\in B,\hfill \\ \hfill & -\Delta v+v={\lambda }_2{\left|x\right|}^{\alpha }{v}^3+\mu {\left|x\right|}^{\beta }{u}^{2}v,x\in B,\hfill \\ \hfill & u,v>0,x\in B,u=v=0,x\in \partial B,\hfill \end{array}\right.$where $B\subset {\mathbb{R}}^N(N=2,3)$ is the unit ball centered at the origin, ${\lambda }_1,{\lambda }_2>0$, $\mu >0$, $\beta >0$, $\alpha >0$. By virtue of variational approaches and rescaling methods, the system has a nontrivial ground state solution with $\alpha >\beta >0$, moreover, by reduction methods, the ground state solution is radial symmetry if $\beta >0$ small enough.     

Infinitely many solutions for a gauged nonlinear Schrödinger equation

This paper is concerned with a gauged nonlinear Schrödinger equation $-\Delta u+\omega u+\left(\frac{h^2\left(\right|x\left|\right)}{|x{|}^2}+{\int }_{\left|x\right|}^{\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u=f\left(\right|x|,u)in{\mathbb{R}}^2.$Under some suitable conditions on the nonlinearity $f$, we obtain two new existence results of infinitely many high energy solutions by using variational methods, and our results generalize and improve the recent result in the literature.     

Summand absorbing submodules of a module over a semiring

An R-module V over a semiring R lacks zero sums (LZS) if $x+y=0$ implies $x=y=0$. More generally, we call a submodule W of V “summand absorbing” (SA) in V if $?x,y\in V:x+y\in W?x\in W,y\in W$. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.