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Vol 13 - Issue 3 (June 2022)

Advances in Pure and Applied Mathematics

List of Articles

Limit sets and global dynamic for 2-D divergence-free vector fields

The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and $$${V}$$$ is a divergence-free $$$C^{1}$$$-vector field with finitely many singularities on M then every orbit L of $$$\mathcal{V}$$$ is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M* = M - Sing($$$\mathcal{V}$$$), (iv) a locally dense orbit, where Sing($$$\mathcal{V}$$$) denotes the set of singular points of $$$\mathcal{V}$$$. On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M*. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431-445] established when the divergence-free vector field $$$\mathcal{V}$$$ is regular that is all its singular points are non-degenerate.

Weighted estimates for operators associated to the Bergman-Besov kernels

We characterize the weights for which we have the boundedness of standard weighted integral operators induced by the Bergman-Besov kernels acting between two weighted Lebesgue classes on the unit ball of ℂN in terms of Békollè - Bonami type condition on the weights. To accomplish this we employ the proof strategy originated by Békollè.

A result on Bruck Conjecture related to Shift Polynomials

This paper mainly concerns about establishing the Bruck conjecture for differential-difference polynomial generated by an entire function. The polynomial considered is of finite order and involves the entire function $$$f(z)$$$ and its shift $$$f(z + c)$$$ where $$$c \in ℂ$$$. Suitable examples are given to prove the sharpness of sharing exceptional values of Borel and Nevanlinna.

Existence Results for Singular p(x)-Laplacian Equation

This paper is concerned with the existence of solutions for the following class of singular fourth order elliptic equations
$$$ \left\{ \begin{array}{ll} \Delta\Big(|x|^{p(x)}|\Delta u|^{p(x)-2}\Delta u\Big)=a(x)u^{-\gamma (x)}+\lambda f(x,u),\quad \mbox{in }\Omega, \\ u=\Delta u=0, \quad \mbox{on }\partial\Omega. \end{array} \right.$$$
where $$$\Omega$$$ is a smooth bounded domain in $$$\mathbb{R}^N, \gamma :\overline{\Omega}\rightarrow (0,1)$$$ be a continuous function, $$$f\in C^{1}( \overline{\Omega}\times \mathbb{R}), p:\; \overline{\Omega}\longrightarrow \;(1,\infty)$$$ and $$$a$$$ is a function that is almost everywhere positive in $$$\Omega$$$. Using variational techniques combined with the theory of the generalized Lebesgue-Sobolev spaces, we prove the existence at least one nontrivial weak solution.