A note on uniqueness of the ground state for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation $-{\epsilon }^2\Delta u+V\left(x\right)u={\left|u\right|}^{p-2}u,u\in H^1\left({\mathbb{R}}^N\right),$where $N\ge 3$, $2<p<\frac{2N}{N-2}$. For $\epsilon$ small enough and a class of $V\left(x\right)$, we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of $V\left(x\right)$ and its first derivatives. Here our results are suitable for a kind of $V\left(x\right)$ which has different increasing rates at different directions.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

We consider the following Schrödinger equation $-{\hslash }^2\Delta u+V\left(x\right)u=\Gamma \left(x\right)f\left(u\right)\mathrm{in}{\mathbb{R}}^N,$where $u\in H^1\left({\mathbb{R}}^N\right)$, $u>0$, $\hslash >0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $\Gamma$ attains global maximum at the same point or $V$ attains global minimum and $\Gamma$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash >0$.     

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.     

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).     

Global existence and decay estimate of solutions to magneto-micropolar fluid equations

We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in ${\mathbb{R}}^3$. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms ${\dot{H}}^{?s}$ $(0\le s<\frac{3}{2})$ or homogeneous Besov norms ${\dot{B}}_{2,\infty }^{?s}$ $(0<s\le \frac{3}{2})$, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual $L^p?L^2$ $(1\le p\le 2)$ type of the decay rates without requiring that the $L^p$ norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in ${\mathbb{R}}^2$ with large initial data.     

Matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. We use $E_3(2d?1,n?2d+1)$ to denote the 3-uniform hypergraph whose vertex set can be partitioned into two vertex classes $V_1$ and $V_2$ of size $2d?1$ and $n?2d+1$, respectively, and whose edge set consists of all the triples containing at least two vertices of $V_1$. Let $H$ be a 3-uniform hypergraph of order $n\ge 13d$ with no isolated vertex and $\text{deg}\left(u\right)+\text{deg}\left(v\right)>2\left(\left(\genfrac{}{}{0ex}{}{n?1}{2}\right)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)\right)$ for any two adjacent vertices $u,v\in V\left(H\right)$. In this paper, we show that $H$ contains a matching of size $d$ if and only if $H$ is not a subgraph of $E_3(2d?1,n?2d+1)$. This result improves our previous one in Zhang and Lu (2018).     

On three color Ramsey numbers R(C4,C4,K1,n)

For $k$ given graphs $G_1,G_2,\dots ,G_k$, $k\ge 2$, the $k$-color Ramsey number, denoted by $R(G_1,G_2,\dots ,G_k)$, is the smallest integer $N$ such that if we arbitrarily color the edges of a complete graph of order $N$ with $k$ colors, then it always contains a monochromatic copy of $G_i$ colored with $i$, for some $1\le i\le k$. Let $C_m$ be a cycle of length $m$ and $K_{1,n}$ a star of order $n+1$. In this paper, firstly we give a general upper bound of $R(C_4,C_4,\dots ,C_4,K_{1,n})$. In particular, for the 3-color case, we have $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+3$ and this bound is tight in some sense. Furthermore, we prove that $R(C_4,C_4,K_{1,n})\le n+?\sqrt{4n+5}?+2$ for all $n={?}^2??$ and $?\ge 2$, and if $?$ is a prime power, then the equality holds.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

Erdős–Gallai stability theorem for linear forests

The Erd?s–Gallai Theorem states that every graph of average degree more than $l?2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=\left({?}_{i=1}^k{P}_{2a_i}\right)?\left({?}_{i=1}^l{P}_{2b_i+1}\right)$ be $k+l$ disjoint paths of order $2a_1,\dots ,2a_k,2b_1+1,\dots ,2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\ge 2$. If the minimum degree $\delta \left(G\right)\ge {\sum }_{i=1}^k{a}_i+{\sum }_{i=1}^l{b}_i?1$, then $F?G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

Energy non-collapsing and refined blowup for a semilinear heat equation

Refined structures of blowup for non-collapsing maximal solution to a semilinear parabolic equation

$u_t?△u=|u{|}^{p?1}u$

with $p>1$ are studied. We will prove that the blowup set is empty for non-collapsing blowing-up in subcritical case, and all finite time non-collapsing blowing-up must be refined type II in critical case. When $p>p_S\equiv \frac{N+2}{N?2}$ for $N\ge 3$, the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than $N?2?\frac{4}{p?1}$, which answers a question proposed in [7] positively. At the end of this paper, we also present some new examples of collapsing and non-collapsing blowups.     

The covering radius of the Reed–Muller code RM(2,7) is 40

It was proved by J. Schatz that the covering radius of the second order Reed–Muller code $RM(2,6)$ is 18 (Schatz (1981)). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for the covering radius of $RM(2,n)$, $n=8,9,10$.     

Sharp upper bounds of the spectral radius of a graph

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges. The spectral radius $\rho \left(G\right)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of $\rho \left(G\right)$ which improves some known bounds: If $\frac{(k?2)(k?3)}{2}\le m?n\le \frac{k(k?3)}{2}$, where $k(3\le k\le n)$ is an integer, then $\rho \left(G\right)\le \sqrt{2m?n?k+\frac{5}{2}+\sqrt{2m?2n+\frac{9}{4}}}.$The equality holds if and only if $G$ is a complete graph $K_n$ or $K_4?e$, where $K_4?e$ is the graph obtained from $K_4$ by deleting some edge $e$.