Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in $R^{n}\backslash\{0\}$

We consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

There Is Only One Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

Solution of Integral Equations by Dunkl and Distributional Dunkl Transform

The paper investigates the Dunkl transform and distributional Dunkl transform and the basic properties as convolution. The integral equations such as Volterra integral equation of first and second kind and Abel integral equation are solved by using dunkl transform. Further, solution obtained is considered in distributional sense by employing integral equations to distribution spaces and as well as using the distributional Dunkl transform for its solution.

There Is Only One Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions ofLp-Growth

LetCbe a regular cone inℝand letTC=ℝ+iC⊂ℂbe a tubular radial domain. LetUbe the convolutor in Beurling ultradistributions ofLp-growth corresponding toTC. We define the Cauchy and Poisson integral ofUand show that the Cauchy integral of Uis analytic inTCand satisfies a growth property. We represent Uas the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation ofU. Also we show that the Poisson integral ofUcorresponding toTCattainsUas boundary value in the distributional sense.

CHARACTERIZATION OF POLYNOMIALS

In 1954, it was proved that if f is infinitely differentiable in the interval I and some derivatives (of order depending on x) vanish at each x, then f is a polynomial. Later, it was generalized for the multivariable case. A further extension for distributions is possible. If Ω ⊆ Rn is a non-empty connected open set, [Formula: see text] and for every [Formula: see text], there exists m(φ) ∈ N such that (Dαu)(φ) = 0 for all multi-indices α satisfying ‖α‖ = m(φ), then u is a polynomial (in distributional sense).

ON THE RECOVERY OF A MANIFOLD WITH PRESCRIBED METRIC TENSOR

A classical theorem in differential geometry asserts that if a C2-metric tensor satisfies the Riemann compatibility conditions, then it is induced by an immersion. We prove that this theorem still holds true for C1-metric tensors satisfying the Riemann compatibility conditions in a distributional sense.

New inversion formulas for the Krätzel transformation

We study in distributional sense by means of the kernel method an integral transform introduced by Krätzel. It is well known that the cited transform generalizes to the Laplace and Meijer transformation. Properties of analyticity, boundedness, and inversion theorems are established for the generalized transformation.