Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

A scheme for interpolation by Hankel translates of a basis function

This work discusses interpolation of complex-valued functions defined on the positive real axis $I$ by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415–450]. The set of interpolation points will be a subset $\{a_1,\dots ,a_n\}$ of $I$ and the interpolants will take the form $u\left(x\right)=\sum _{i=1}^n{\alpha }_i\left({\tau }_{a_i}?\right)\left(x\right)+\sum _{j=0}^{m?1}{\beta }_j{p}_{\mu ,j}\left(x\right)(x\in I)\text{,}$ where $\mu \ge ?1/2,?$ is a complex function defined on $I$ (the so-called basis function), $p_{\mu ,j}\left(x\right)=x^{2j+\mu +1/2}(j\in {\mathbb{Z}}_{+},0\le j\le m?1)$ is a Müntz monomial, ${\tau }_z(z\in I)$ denotes the Hankel translation operator of order $\mu$, and ${\alpha }_i,{\beta }_j(i,j\in {\mathbb{Z}}_{+},1\le i\le n,0\le j\le m?1)$ are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN

In the present paper, the following Schrödinger–Kirchhoff-type problem: (1.1)$?(a+b{\int }_{R^N}{\left|?u\right|}^{2}dx)\Delta u+V\left(x\right)u=f(x,u)\text{,}\text{in}{R}^N$ is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem.