Subcohomology for Zooming Maps

In the context of zooming maps $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, if the map is topologically exact, we prove that a potential $\phi : M \to \mathbb{R}$ for which the integrals $\int \phi d\mu \geq 0$ with respect to any $f$-invariant probability $\mu$, admits a coboundary $\lambda_{0}- \lambda_{0} \circ T$ such that $\phi \geq \lambda_{0}- \lambda_{0} \circ T$. This extends a result for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context.Comment: This new version contains substantial changes, as a new proof for the main theorem. arXiv admin note: substantial text overlap with arXiv:2010.0814

A new class of multiple nonlocal problems with two parameters and variable-order fractional p(⋅)p(\cdot)-Laplacian

In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $p(x)$-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(\sigma_{p(x,y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) =\lambda |u|^{q(x)-2}u\left(\int_\O\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_\O\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }\Omega, \\ u=0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*} where the nonlocal term is defined as $$\sigma_{p(x,y)}(u)=\int_{\Omega\times \Omega}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}} \,dx\,dy.$$ Here, $\Omega\subset\mathbb{R}^{N}$ represents a bounded smooth domain with at least $N\geq2$. The function $M(s)$ is given by $M(s) = a - bs^\gamma$, where $a\geq 0$, $b>0$, and $\gamma>0$. The parameters $k_1$, $k_2$, $\lambda$ and $\beta$ are real parameters, while the variables $p(x)$, $s(\cdot)$, $q(x)$, and $r(x)$ are continuous and can change with respect to $x$. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when $a>0$ and when $a=0$. To the best of our knowledge, these results are the first contributions to research on the variable-order $p(x)$-fractional Laplacian operator.Comment: 21 page

A degenerate Kirchhoff-type problem involving variable s(⋅)s(\cdot)-order fractional p(⋅)p(\cdot)-Laplacian with weights

This paper deals with a class of nonlocal variable $s(.)$-order fractional $p(.)$-Kirchhoff type equations: \begin{eqnarray*} \left\{ \begin{array}{ll} \mathcal{K}\left(\int_{\mathbb{R}^{2N}}\frac{1}{p(x,y)}\frac{|\varphi(x)-\varphi(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right)(-\Delta)^{s(\cdot)}_{p(\cdot)}\varphi(x) =f(x,\varphi) \quad \mbox{in }\Omega, \\ \varphi=0 \quad \mbox{on }\mathbb{R}^N\backslash\Omega. \end{array} \right. \end{eqnarray*} Under some suitable conditions on the functions $p,s, \mathcal{K}$ and $f$, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the $p(\cdot)$ fractional setting

[Generalized Telegraph equation with fractional p(x)p(x)-Laplacian

The purpose of this paper is devoted to \textcolor{red}{discussing} the existence of solutions for a generalized fractional telegraph equation involving a class of $\psi$-Hilfer fractional with $p(x)$-Laplacian differential equation.Comment: 14 page

Existence and multiplicity of solutions involving the p(x)p(x)-Laplacian equations

We study a class of $p(x)$-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland\u27s variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke\u27s theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results