Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality _Δφu?f(u)?(|_∇0u|)${\mathrm{\Delta }}_{\phi }u?f(u)?(|{\mathrm{\nabla }}_{0}u|)$ on the Heisenberg group under a generalized Keller–Osserman condition. The operator _Δφu${\mathrm{\Delta }}_{\phi }u$ is the φ  -Laplacian defined by _div0(|_∇0u|⁻¹φ(|_∇0u|)_∇0u)${\mathrm{div}}_0({|{\mathrm{\nabla }}_{0}u|}^{-1}\phi (|{\mathrm{\nabla }}_{0}u|){\mathrm{\nabla }}_{0}u)$ and φ, f and ? satisfy mild structural conditions. In particular, ? is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.     

Soliton dynamics for the generalized Choquard equation

We investigate the soliton dynamics for a class of nonlinear Schrödinger equations with a non-local nonlinear term. In particular, we consider what we call generalized Choquard equation   where the nonlinear term is (|x|^θ−N?|u|^p)|u|^p−2u$({|x|}^{\theta -N}?{|u|}^p){|u|}^{p-2}u$. This problem is particularly interesting because the ground state solutions are not known to be unique or non-degenerate.     

Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)u^p=0$\mathrm{\Delta }u+K(|x|)u^p=0$ in R^N${\mathbf{R}}^N$ for N>2$N>2$ and p>1$p>1$, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r^−?K(r)$r^{-?}K(r)$ with ?>−2$?>-2$ around a positive constant is small near r=∞$r=\infty$ and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent _pc$p_c$ which is determined by N   and the order of the behavior of K(r)$K(r)$ as r=|x|→0$r=|x|\to 0$ and ∞. In order to understand how subtle the structure is on K   at p=_pc$p=p_c$, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)$p=(N+2)/(N-2)$.     

On spaces of modular forms spanned by eta-quotients

An eta-quotient of level N   is a modular form of the shape f(z)=_∏δ|Nη(δz)^_rδ$f(z)={\prod }_{\delta |N}\eta {(\delta z)}^{r_{\delta }}$. We study the problem of determining levels N   for which the graded ring of holomorphic modular forms for _Γ0(N)${\Gamma }_0(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N  . In addition, we prove that if f(z)$f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then f(z)$f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of _J0(2^k)(Q)$J_0(2^k)(\mathbb{Q})$.     

A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let Ω   be a smooth bounded simply connected domain in R²${\mathbb{R}}^2$. We investigate the existence of critical points of the energy _Eε(u)=1/2_∫Ω|∇u|²+1/(4ε²)_∫Ω(1−|u|²)²$E_{\epsilon }(u)=1/2{\int }_{\Omega }{|\mathrm{\nabla }u|}^2+1/(4{\epsilon }^2){\int }_{\Omega }{(1-{|u|}^2)}^2$, where the complex map u has modulus one and prescribed degree d on the boundary. Under suitable nondegeneracy assumptions on Ω, we prove existence of critical points for small ε. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disk. Next, we prove that critical points exist in “most” of the domains.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

Traveling wave solutions of the heat flow of director fields

We consider the simplest possible heat equation for director fields, ut=Δu+|∇u|²u$u_t=\mathrm{\Delta }u+{|\mathrm{\nabla }u|}^{2}u$ (|u|=1$|u|=1$), and construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1.