The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity ${\mathrm{e}}^{\frac{2\pi \mathbf{i}}{k}}$ (and its symmetric entry is ${\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set $\{1,{\mathrm{e}}^{\frac{2\mathit{\pi }\mathbf{i}}{k}},{\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}\}$, which is not closed under multiplication when $k?4$. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank ($H_k$-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with $H_k$-rank 2 and 3, respectively.     

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations$\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u?\mathrm{\Delta }u+V(x)u?\frac{1}{2}u\mathrm{\Delta }(u^2)=g(u),\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2({\mathbb{R}}^N)\hfill \end{array}$ where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition $g(t)t\ge \mu G(t)\ge 0$ with $\mu >3$. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.     

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.     

Exact decay estimate on solutions of critical p-Laplacian equations in R+N

Consider the critical p-Laplacian equation in ${\mathbb{R}}_{+}^N$

$\{\begin{array}{cc}\hfill & ?{\mathrm{\Delta }}_{p}u=0,\text{in}{\mathbb{R}}_{+}^N,\hfill \\ \hfill & |\mathrm{?}u{|}^{p?2}\frac{?u}{?n}=|u{|}^{\stackrel{￣}{p}?2}u,\text{on}{\mathbb{R}}^{N?1},\hfill \end{array}$

where ${\mathbb{R}}_{+}^N={\mathbb{R}}^{N?1}\times (0,\infty )$, $1<p<N,\stackrel{￣}{p}=\frac{(N?1)p}{N?p}$ is the critical exponent of the Sobolev imbedding from $W^{1,p}({\mathbb{R}}_{+}^N)$ to $L^q({\mathbb{R}}^{N?1})$, $p\le q\le \stackrel{￣}{p}$ and ${\mathrm{\Delta }}_p$ is the p-Laplacian operator, ${\mathrm{\Delta }}_{p}u=\mathrm{?}(|\mathrm{?}u{|}^{p?2}\mathrm{?}u)$. We prove polynomial decay of the solutions

$|u(x)|\le c{(1+|x|)}^{?\frac{N?p}{p?1}},\text{for}x\in {\mathbb{R}}_{+}^N.$

The decay exponent is the best possible.     

Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

We consider the nonlinear problem of inhomogeneous Allen–Cahn equation

${?}^2\mathrm{\Delta }u+V(y)u(1?u^2)=0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{on}?\mathrm{\Omega },$

where Ω is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, ? is a small positive parameter, ν denotes the unit outward normal of ?Ω, V is a positive smooth function on $\overline{\mathrm{\Omega }}$. Let Γ be a curve intersecting orthogonally with ?Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ${\int }_{\mathrm{\Gamma }}{V}^{1/2}$. We can prove that there exists a solution $u_{?}$ such that: as $?\to 0$, $u_{?}$ approaches +1 in one part of Ω, while tends to ?1 in the other part, except a small neighborhood of Γ.     

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

This paper discusses the quasilinear Schrödinger equation $-\Delta u+V\left(x\right)u-\Delta \left[{(1+u^2)}^{\frac{1}{2}}\right]\frac{u}{2{(1+u^2)}^{\frac{1}{2}}}=K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N,$where $N⩾3$. Under appropriate assumptions on the potentials $V$ and $K$ and local sublinear growth assumptions on the nonlinear term $f$, we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.     

Solutions for conformally invariant fractional Laplacian equations with multi-bumps centered in lattices

In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:

${(?\mathrm{\Delta })}^{s}u=K(x)u^{\frac{N+2s}{N?2s}},u>0\mathrm{in}{\mathbb{R}}^N,$

where $s\in (0,1)$ and $N>2+2s$, $K>0$ is periodic in $(x_1,\dots ,x_k)$ with $1\le k<\frac{N?2s}{2}$. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in ${\mathbb{R}}^k$, including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in ${\mathbb{R}}^k$, the restriction on $1\le k<\frac{N?2s}{2}$ is in some sense optimal, since we can show that for $k\ge \frac{N?2s}{2}$, no such solutions exist.     

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).     

Existence of multi-bump solutions for a system with critical exponent in RN

We consider the following system with critical exponent in ${\mathbb{R}}^N$:$\{\begin{array}{c}?\mathrm{\Delta }u=K_1(y)u^{2^{?}?1}+\frac{p}{2^{?}}V(y)u^{p?1}{v}^q\text{in}{\mathbb{R}}^N,\hfill \\ ?\mathrm{\Delta }v=K_2(y)v^{2^{?}?1}+\frac{q}{2^{?}}V(y)u^p{v}^{q?1}\text{in}{\mathbb{R}}^N,\hfill \\ u,v>0,y\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 5$, $p,q>1$ and $p+q=2^{?}=\frac{2N}{N?2}$. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in ${\mathbb{R}}^N$. Since $p<2$ or $q<2$, we introduce two new norms to avoid singularity.     

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Let $\mathrm{\Omega }?{\mathbb{R}}^N$ ($N\ge 3$) be a bounded $C^2$ domain and $\delta (x)=\mathrm{dist}(x,?\mathrm{\Omega })$. Put $L_{\mu }=\mathrm{\Delta }+\frac{\mu }{{\delta }^2}$ with $\mu >0$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to

$?L_{\mu }u=u^p+\tau \text{in }\mathrm{\Omega },u=\nu \text{on }?\mathrm{\Omega },$

where $\mu >0$, $p>0$, τ and ν are measures on Ω and ?Ω respectively. We then establish existence results for the system

$\{\begin{array}{cc}\hfill & ?L_{\mu }u=?v^p+\tau \text{in }\mathrm{\Omega },\hfill \\ \hfill & ?L_{\mu }v=?u^{\stackrel{?}{p}}+\stackrel{?}{\tau }\text{in }\mathrm{\Omega },\hfill \\ \hfill & u=\nu ,v=\stackrel{?}{\nu }\text{on }?\mathrm{\Omega },\hfill \end{array}$

where $?=\pm 1$, $p>0$, $\stackrel{?}{p}>0$, τ and $\stackrel{?}{\tau }$ are measures on Ω, ν and $\stackrel{?}{\nu }$ are measures on ?Ω. We also deal with elliptic systems where the nonlinearities are more general.     

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type $y^2=x^3+f(t)x$, defined over the finite field ${\mathbb{F}}_q$ for a prime power $q\equiv 3\mathrm{mod}4$. It is shown that if $s^4=f(t)$ defines a curve that is maximal over ${\mathbb{F}}_{q^2}$ then the rank of the group of sections defined over ${\mathbb{F}}_q$ on the elliptic surface is determined in terms of elementary properties of the rational function $f(t)$. Similar results are shown for elliptic surfaces given by $y^2=x^3+g(t)$ using prime powers $q\equiv 5\mathrm{mod}6$ and curves $s^6=g(t)$. Finally, for each of the forms used here, existence of curves with the property that they are maximal over ${\mathbb{F}}_{q^2}$ is discussed, as well as various examples.     

Higher dimensional Ginzburg–Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<?\le 1$, let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded, simply connected, smooth domain, and $u_{?}:\mathrm{\Omega }?{\mathbb{R}}^n\to {\mathbb{R}}^2$ solve the Ginzburg–Landau equation under the weak anchoring boundary condition:

$\{\begin{array}{cc}?\mathrm{\Delta }{u}_{?}=\frac{1}{{?}^2}(1?|u_{?}{|}^2)u_{?}\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ \frac{?u_{?}}{?\nu }+{\lambda }_{?}(u_{?}?g_{?})=0\hfill & \mathrm{on}?\mathrm{\Omega },\hfill \end{array}$

where the anchoring strength parameter ${\lambda }_{?}=K{?}^{?\alpha }$ for some $K>0$ and $\alpha \in [0,1)$, and $g_{?}\in C^2(?\mathrm{\Omega },{\mathbb{S}}^1)$. Motivated by the connection with the Landau–De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the asymptotic behavior of $u_{?}$ as ? goes to zero under the condition that the total modified Ginzburg–Landau energy satisfies $F_{?}(u_{?},\mathrm{\Omega })\le M|\log ??|$ for some $M>0$.