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.     

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.     

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 $f$.     

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.     

Existence of multi-solitary waves with logarithmic relative distances for the NLS equation

We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,

$\mathrm{i}{?}_{t}u+\mathrm{\Delta }u+|u{|}^{p?1}u=0,t\in \mathbb{R},x\in {\mathbb{R}}^d,$

in mass-subcritical cases $1<p<1+\frac{4}{d}$ and mass-supercritical cases $1+\frac{4}{d}<p<\frac{d+2}{d?2}$, i.e. solutions $u(t)$ satisfying

${6u(t)?{\mathrm{e}}^{\mathrm{i}\gamma (t)}\sum _{k=1}^{2}Q(??x_k(t))6}_{H^1}\to 0$

and

$|x_1(t)?x_2(t)|～2\log ?t,\text{as}t\to +\infty ,$

where Q is the ground state. The logarithmic distance is related to strong interactions between solitary waves.In the integrable case ($d=1$ and $p=3$), the existence of such solutions is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case $p=1+\frac{4}{d}$ exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).