In this paper we study the decay property of solutions of a semilinear Schrödinger equation, $-(∇ - iA)^{2}u+(V-E)u=Q|u|^{p-2}u$, on $\mathbb R^{n}$, where $n \geq 2$ and $2<p <2^{*}$. We give a lower bound estimate of nontrivial solutions at infinity. In two-dimensional case, we give super-exponential decay estimates of solutions at infinity. Moreover, we show the existence of a nontrivial solution under additional assumptions on potentials.

The purpose of this note is to establish some uniqueness results for hemivariational inequality.

We are concerned with the type problem for the covering surface $S_{\Gamma}$ or more precisely $(S_{\Gamma},S,\pi_{\Gamma})$ of the base parabolic Riemann surface $S$ with its projection $\pi_{\Gamma}$, where $S_{\Gamma}$ is the infinitely sheeted covering Riemann surface constructed from the sequence of replicas $S_{n}$ of $S$ and the family $\Gamma=\{\gamma_{n}:n \in \Bbb N \}$ of pasting arcs $\gamma_{n} \subset S$ with $\gamma_{n-1} \cap \gamma_{n}=\emptyset\(n \in \Bbb N)$ in such a fashion that $S_{n}\setminus(\gamma_{n-1}\cup\gamma_{n})$ is joined to $S_{n+1}\setminus(\gamma_{n}\cup\gamma_{n+1})$ crosswise along $\gamma_{n}$ for each $n \in \Bbb N$. Here $\Bbb N$ is the set of positive integers and the parabolicity of a surface is characterized by the nonexistence of the Green function on the surface. The central object of this paper is to show by using the theory of Wiener and Royden compactifications that the type of the covering surface $S_{\Gamma}$ is parabolic if the sequence of capacities ${\rm cap}(\gamma_{n},S\setminus\gamma_{0})$ of $\gamma_{n}\in\Gamma$ with respect to the surface $S$ less the arc $\gamma_{0}$ fixed in $S$ disjoint from all other arcs $\gamma_{n}\in\Gamma$ for all sufficiently large $n\in\Bbb N$ converges to zero so rapidly as to satisfy the condition $\sum_{n\in\Bbb N}{\rm cap} (\gamma_{n}, S\setminus\gamma_{0})<+\infty$.

We provide estimates on the degree of $C^{l}-\mathcal{G}_{V}$-determinacy ($\mathcal{G}$ is one of Mather's groups $\mathcal{R} $ or $\mathcal{K}$) of weighted homogeneous function germs which are defined on weighted homogeneous analytic variety $V$ and satisfies a convenient Lojasiewicz condition. The result gives an explicit order such that the $C^{l}$-geometrical structure of a weighted homogeneous polynomial function germ is preserved after higher order perturbations, which generalize the result on $C^{l}-\mathcal{K}$-determinacy of weighted homogeneous functions germs given by M. A. S. Ruas.

We provide estimates for the $C^{\ell}$-$G$-triviality, for $ 0 \leq \ell < \infty$ and $G$ is one of Mather's groups ${\mathcal R}$, ${\mathcal C}$ or ${\mathcal K}$, of deformations of analytic map germs $f: (\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ of type $f_t(x)=f(x)+θ(x,t)$ which satisfy a non-degeneracy condition with respect to some Newton polyhedron. We apply the method of construction of controlled vector fields and, for each group $G$, the control function is determined from the choice of a convenient {\it Newton filtration } in the ring of real analytic germs. The results are given in terms of the filtration of the coordinate function germs $f_1, \ldots , f_p$ of $f$.

We present two comparison theorems for a class of weakly coupled systems of Hamilton-Jacobi equations with convex, coercive Hamiltonians. These results apply in particular to systems arising in large deviations theory for random evolution processes.

The zero modes and zero resonances of the Dirac operator $H=\alpha\cdot D + Q(x)$ are discussed, where $\alpha= (\alpha_1, \, \alpha_2, \, \alpha_3)$ is the triple of $4 \times 4$ Dirac matrices, $D=\frac{1}{i} ∇_x$, and $Q(x)=( q_{jk} (x))$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C \langle x \rangle^{-\rho}$, $\rho >1$. We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $ρ > 3/2$.

We show that the Cauchy integral operator is bounded from $H^p(R^1)$ to $h^p(R^1)$ (local Hardy space). To prove our theorem we shall introduce generalized atom and consider a variant of "$Tb$ theorem".

We study a class of Butler groups of infinite rank, called Hawaiian groups.They are defined as subgroups of a rational vector space and contain parameters that provide for flexibility but are concrete enough to allow for the computation of certain crucial subgroups and quotient groups, to exhibit endomorphisms and describe the endomorphism rings. Most Hawaiian groups are finitely Butler; under stronger assumptions they are not finitely filtered and hence not $B_2$-groups.