A theorem on computation of the homological Conley index of an isolated invariant set of the Poincaré map associated to a section in a rotating local dynamical system $\phi$ is proved. Let $(N,L)$ be an index pair for a discretization $\phi^h$ of $\phi$, where $h> 0$, and let $S$ denote the invariant part of $N\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, the cycles $u_j$ and $\sum a_{ij}u_i$ are homologous in the covering pair $\big(\widetilde N,\widetilde L\big)$ of $(N,L)$ and the homology relation is preserved in $\big(\widetilde N,\widetilde L\big)$ under the transformation induced by $\phi^t$ for $t\in [0,h]$ then the homological Conley index of $S_0$ is equal to the Leray reduction of the matrix $[a_{ij}]$. In particular, no information on the values of the Poincaré map or its approximations is required. In a special case of the system generated by a $T$-periodic non-autonomous ordinary differential equation with rational $T/h> 1$, the theorem was proved in the paper M. Mrozek, R. Srzednicki, and F. Weilandt, SIAM J. Appl. Dyn. Syst. 14 (2015), 1348-1386, and it motivated a construction of an algorithm for determining the index.

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight$$\begin{cases}-\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\ u> 0& \text{in } \Omega,\\[1mm]\dfrac{\partial u}{\partial\nu}=0 & \text{on } \partial\Omega,\end{cases}$$where $\nu$ denotes the outer unit normal vector to $\partial\Omega$, $q\in\overline{\Omega}$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p> 1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions.We show that if $q\in\Omega$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while, if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\nu(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overline{\Omega}$ is an isolated local maximum point of $a(x)$ or not.

We study the following nonlinear eigenvalue problem$$-u''(t) = \lambda u(t)^p\log(1+u(t)), \quad u(t) > 0,\quad t \in I := (-1,1), \quad u(\pm 1) = 0,$$where $p \ge 0$ is a given constant and $\lambda > 0$ is a parameter. It is known that, for any given $\alpha > 0$, there exists a unique classical solution pair $(\lambda(\alpha), u_\alpha)$ with $\alpha = \Vert u_\alpha\Vert_\infty$. We establish the asymptotic formulas for the bifurcation curves $\lambda(\alpha)$ and the shape of solution $u_\alpha$ as $\alpha \to \infty$ and $\alpha \to 0$.

In this paper, we are concerned with the existence of sign-changing multi-bump solutions for the following nonlinear Choquard equation\begin{equation}\tag{0.1}-\Delta u+(\lambda V(x)+1)u=(I_{\alpha}\ast|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N,\end{equation}where $I_\alpha$ is the Riesz potential, $\lambda \in \mathbb{R}^{+}$, $ (N-4)^{+}< \alpha< N$,$2\le p < ({N+\alpha})/({N-2})$, and $V(x)$ is a nonnegative continuous function with a potential well $\Omega:= {\rm int}(V^{-1}(0))$ which possesses $k$ disjoint bounded components $\Omega_1, \ldots, \Omega_k$. We prove the existence of sign-changing multi-bump solutions for (0.1) if $\lambda$ is large enough.

In this paper, we study and establish a positive answer to a long-standing open problem raised by A.T.-M. Lau in 1976. It is about whether the left amenability property of the Banach algebra WAP($S$), of all weakly almost periodic functions, on a given semitopological semigroup $S$ is equivalent to the existence of a common fixed point of any separately weakly continuous and weakly quasi-equicontinuous nonexpansive action of $S$ on a nonempty weakly compact convex subset of a separated locally convex space. We establish here an affirmative answer; and among other things, we show that the affine counterpart of this question holds and also the AP($S$) formulation of this problem is true.

In this paper, we consider the existence of positive solutions with prescribed normalizations for strongly coupled fractional Kirchhoff type systems in the whole space $\mathbb{R}^N$ $(N=2,3)$. Under constant vanishing potentials and attractive interspecies interactions, two cases are studied: one is $L^2$-subcritical and the other is $L^2$-supercritical. In the first case, we prove the existence of a positive solution by the constrained minimizing methods. In the second case, by using a minimax procedure, we prove the existence of a mountain pass type solution under high perturbations of the coupling parameter, which is also a ground state solution. Moreover, we study the $L^2$-critical case under certain type of trapping potentials. In this case, we are concerned with not only attractive but also repulsive interspecies interactions, and prove the existence of a positive solution by introducing some auxiliary minimization problems. These conclusions extend some known ones in previous papers.

We present a short proof of S. Parsa's theorem that there exists a compact $n$-polyhedron $P$, $n\ge 2$, non-embeddable in $\mathbb R^{2n}$, such that $P*P$ embeds in $\mathbb R^{4n+2}$. This proof can serve as a showcase for the use of geometric cohomology. We also show that a compact $n$-polyhedron $X$ embeds in $\mathbb R^m$, $m\ge {3(n+1)}/2$, if either

$\bullet$ $X*K$ embeds in $\mathbb R^{m+2k}$, where $K$ is the $(k-1)$-skeleton of the $2k$-simplex; or

$\bullet$ $X*L$ embeds in $\mathbb R^{m+2k}$, where $L$ is the join of $k$ copies of the $3$-point set; or

$\bullet$ $X$ is acyclic and $X\times{\rm (triod)}^k$ embeds in $\mathbb R^{m+2k}$.

In this paper, we are concerned with the following Kirchhoff equations involving the fractional Laplacian,\begin{equation}\tag{0.1}\begin{cases}\left(a+b[u]^2_s\right)(-\Delta)^su=u^p+h(x,u,\nabla u), &x\in\Omega,\\u> 0, &x\in\Omega,\\u=0, &x\notin\Omega,\\\end{cases}\end{equation}where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$ $(N\geq3)$, $0< s< 1$, $a,b> 0$ and $0< p< ({N+2s})/({N-2s})$ are constants. Under suitable conditions on $h(x,u,\nabla u)$, using the defining integral, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Moreover, existence results for positive solutions of problem (0.1) are proved by Leray-Schauder degree theory and the above estimates.

In this work, we consider the collection of necessary homological conditions previously obtained via Conley index theory for a Lyapunov semi-graph to be associated to a Gutierrez-Sotomayor flow on an isolating block and address their sufficiency. These singular flows include regular $\mathcal{R}$, cone $\mathcal{C}$, Whitney $\mathcal{W}$, double $\mathcal{D}$ and triple $\mathcal{T}$ crossing singularities. Local sufficiency of these conditions are proved in the case of Lyapunov semi-graphs along with a complete characterization of the branched $1$-manifolds that make up the boundary of the block. As a consequence, global sufficient conditions are determined for Lyapunov graphs labelled with $\mathcal{R}$, $\mathcal{C}$, $\mathcal{W}$, $\mathcal{D}$ and $\mathcal{T}$ and with minimal weights to be associated to Gutierrez-Sotomayor flows on closed singular $2$-manifolds. By removing the minimality condition, we prove other global realizability results by requiring that the Lyapunov graph be labelled with $\mathcal{R}$, $\mathcal{C}$ and $\mathcal{W}$ singularities or that it be linear.

Let $(\Sigma,g)$ be a closed Riemann surface and $H^1(\Sigma)$ be the usual Sobolev space. For any real number $\rho$, we define a generalized mean field type functional $J_{\rho,\phi}\colon H^1(\Sigma)\rightarrow \mathbb{R}$ by\begin{equation*}J_{\rho,\phi}(u)=\frac{1}{2 } \bigg(\int_{\Sigma}|\nabla_g u|^{2} d v_g+\int_{\Sigma}\phi (u-\overline{u}) d v_g \bigg)-\rho\ln \int_{ \Sigma} h e^{u-\overline{u}} d v_g,\end{equation*}where $h\colon \Sigma\to\mathbb{R}$ is a smooth positive function, $\phi\colon \mathbb{R}\to\mathbb{R}$ is a smooth one-variable function and $\overline{u}=\int_\Sigma ud v_g/|\Sigma|$. If $\rho\in (8k\pi,8(k+1)\pi)$ ($k\in \mathbb{N}^{*}$), $\phi$ satisfies $|\phi(t)|\leq C (|t|^p+1)\ (1< p< 2)$ and $|\phi^\prime(t)|\leq C (|t|^{p-1}+1)$ for some constant $C$, then we get critical points of $J_{\rho,\phi}$ by adapting min-max schemes of Ding, Jost, Li and Wang [13], Djadli [14] and Malchiodi [22].

We show that the parametrised topological complexity of Cohen, Farber and Weinberger gives an invariant of group epimorphisms. We extend various bounds for the topological complexity of groups to obtain bounds for the parametrised topological complexity of epimorphisms. Several applications are given, including an alternative computation of the parametrised topological complexity of the planar Fadell-Neuwirth fibrations which avoids calculations involving cup products. We also prove a homotopy invariance result for parametrised topological complexity of fibrations over different bases.

The purpose of this note is to study the equi-continuous non-autonomous dynamical systems. We prove that the topological pressure of the system coincides with the topological pressure restricted on its non-wandering set. To prove this result, due to the lack of an appropriate variational principle for non-autonomous systems, we need to overcome some challenges. We also consider the weakly contractive iterated function systems (IFS), and find that the invariant set of the IFS plays a similar role as the non-wandering set of non-autonomous system.

Given a pair of maps from a two-complex into the circle, for which there exists essential coincidence, we compare the efficiency of three strategies to annihilate all of them, via homotopy deformation. The strategies consist of attaching an arc to the circle in different ways: gluing the arc just by one of its endpoints; gluing the two endpoints to a same point of the circle (so obtaining the eight figure); and gluing the two endpoints to two different points of the circle (so obtaining the theta figure). We prove that the three strategies have the same effect in the matter of annihilate all coincidences, regardless of the domain of the maps. We also study the coincidence problem of maps from closed surfaces into the eight and theta figures, including those one that may not be factored through the circle.

In [1] Barutello, Ortega, and Verzini introduced a non-local functional which regularizes the free fall. This functional has a critical point at infinity and therefore does not satisfy the Palais-Smale condition. In this article we study the $L^2$ gradient flow which gives rise to a non-local heat flow. We construct a rich cascade Morse chain complex which has one generator in each degree $k\ge 1$. Calculation reveals a rather poor Morse homology having just one generator. In particular, there must be a wealth of solutions of the heat flow equation. These can be interpreted as solutions of the Schrödinger equation after a Wick rotation.

In the present work, we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $ \mathbb{R}^2$, for $ a> 0 $, $ \lambda > 0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

We study a classic free boundary problem modeling solid tumor growth. The problem contains a parameter $\mu$. It is well known that the problem admit a unique radially symmetric solution with free boundary $r=R_S$ and a sequence of symmetry-breaking branches of axisymmetric solutions bifurcating from the spherical state $r=R_S$ at an increasing sequence of $\mu= \mu_\ell(R_S)$ ($\ell\geq 2$ even) with free boundary $r= R_S + \varepsilon Y_{\ell,0}(\theta) + O(\varepsilon^2)$, where $Y_{\ell,0}$ is the spherical harmonic of mode $(\ell,0)$. In this paper, we use group-theoretic ideas to obtain a plethora of new branches of non-axisymmetric solutions bifurcating at $\mu= \mu_\ell(R_S)$ $(\ell\geq 2)$. New solutions can model more complex shapes of tumor tissues than the known axisymmetric solutions. The approach is also applicable to many other free boundary problems arising in tumor growth, including a model involving fluid-like tissue.