In this paper we discuss the existence, nonexistence and uniqueness of positive viscosity solution for the following coupled system involving fractional Laplace operator on a smooth bounded domain $\Omega$ in $\mathbb R^n$: \[ \begin{cases} (-\Delta)^{s}u = v^p & \text{in } \Omega,\\ (-\Delta)^{s}v = u^q & \text{in } \Omega,\\ u= v=0 & \text{in } \mathbb R^n\setminus\Omega. \end{cases} \] By means of an appropriate variational framework and a Hölder regularity result for suitable weak solutions of the above system, we prove that such a system admits at least one positive viscosity solution for any $0 < s < 1$, provided that $p, q > 0$, $pq \neq 1$ and the couple $(p,q)$ is below the critical hyperbole \[ \frac{1}{p + 1} + \frac{1}{q + 1} = \frac{n - 2s}{n} \] whenever $n > 2s$. Moreover, by using the maximum principles for the fractional Laplace operator, we show that uniqueness occurs whenever $pq < 1$. Lastly, assuming $\Omega$ is star-shaped, by using a Rellich type variational identity, we prove that no such a solution exists if $(p,q)$ is on or above the critical hyperbole. A crucial point in our proofs is proving, given a critical point $u \in W_{0}^{ s, ({p+1})/{p}}({\Omega}) \cap W^{ 2s, ({p+1})/{p}}(\Omega)$ of a related functional, that there is a function $v$ in an appropriate Sobolev space (Proposition 2.1) so that $(u,v)$ is a weak solution of the above system and a bootstrap argument can be applied successfully in order to establish its H\"{o}lder regularity (Proposition 3.1). The difficulty is caused mainly by the absence of a $L^p$ Calderón-Zygmund theory with $p > 1$ associated to the operator $(-\Delta)^{s}$ for $0 < s < 1$.

Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and strucuture of connecting trajectories of the underlying dynamical system.

To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.

It is shown that any topological action $\Phi$ of a countable orderable and amenable group $G$ on a compact metric space $X$ and every $\Phi$-invariant probability Borel measure $\mu$ admit an extreme partition $\zeta$ of $X$ such that the equivalence relation $R_{\zeta}$ associated with $\zeta$ contains the asymptotic relation $A(\Phi)$ of $\Phi$. As an application of this result and the generalized Glasner theorem it is proved that $A(\Phi)$ is dense for the set $E_{\mu}(\Phi)$ of entropy pairs.

In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.

This paper deals with the existence, uniqueness and regularity of positive $W_{\rm loc}^{1,\Phi}(\Omega)$-solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of the existence is based on a variant of the generalized Galerkin method that we developed inspired by ideas of Browder [4] and a comparison principle. By the use of a kind of Moser's iteration scheme we show the $L^{\infty}(\Omega)$-regularity for positive solutions.

In this paper, we are concerned with the following Kirchhoff type problem with critical growth: \begin{equation*} -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2dx\bigg)\Delta u+V(x)u=f(u)+|u|^4u, \quad u\in H^1(\mathbb R^3), \end{equation*} where $a,b > 0$ are constants. Under certain assumptions on $V$ and $f$, we prove that the above problem has a ground state solution of Nehari-Pohozaev type and a least energy solution via variational methods. Furthermore, we also show that the mountain pass value gives the least energy level for the above problem. Our results improve and extend some recent ones in the literature.

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterizations of a semitopological semigroup $S$ on the existence of a left invariant mean on ${\rm LUC}(S)$, ${\rm AP}(S)$ and ${\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of the Banach algebras related to amenability of groups and semigroups.

The Reidemeister number of an endomorphism of a group is the number of twisted conjugacy classes determined by that endomorphism. The collection of all Reidemeister numbers of all automorphisms of a group $G$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectra of all fundamental groups of solvmanifolds up to Hirsch length 4.

This paper is devoted to the study of a class of singular fractional $p$-Laplacian problems of the form $$ (-\Delta)_p^su-\mu\,\frac{|u|^{p-2}u}{|x|^{ps}} =\alpha\,\frac{|u|^{ p_{s}^{*}(b)-2 }u}{|x|^b} +\beta f(x)|u|^{q-2}u\quad \text{in }\mathbb{R}^N $$ where $0 < s< 1$, $0\leq b < ps < N$, $1< q< p_{s}^{*}(b)$, $\alpha, \beta>0$, $\mu\in \mathbb{R}$, and $f(x)$ is a given function which satisfies some appropriate condition. By using variational methods, we prove the existence of infinitely many solutions under different conditions.

In this paper, we study the existence and nonexistence of positive least energy solutions of the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: \begin{equation*} \begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\left(\frac{1}{|x|^{4}}\ast u^{2}\right)u +\beta \left(\frac{1}{|x|^{4}}\ast v^{2}\right) u, & x \in \Omega,\\ -\Delta v+\nu_{2}v=\mu_{2}\left(\frac{1}{|x|^{4}}\ast v^{2}\right)v +\beta\left(\frac{1}{|x|^{4}}\ast u^{2}\right)v, & x \in \Omega,\\ u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $ (-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.

This article concerns bi-spatial random dynamics for the stochastic reaction-diffusion equation on a thin domain, where the noise is described by a general stochastic process instead of the usual Wiener process. A bi-spatial attractor is obtained when the non-initial state space is the $p$-times Lebesgue space, meanwhile, measurability of the attractor in the Banach space is proved by using measurability of both cocycle and absorbing set. Finally, the $p$-norm convergence of attractors is obtained when the thin domain collapses onto a lower dimensional domain. The method of symbolical truncation is applied to provide some uniformly asymptotic estimates.

We investigate the relationship between the existence of directional derivatives for cone-convex functions with values in a Banach space $Y$ and isomorphisms between $Y$ and $c_0$.

Consider the following elliptic system \begin{equation*} \begin{cases} -\Delta u_i+\mu_i u_i=|u_i|^{2^*-2}u_i+\lambda \sum\limits_{j=1,j\not=i}^ku_j &\text{in }\Omega,\\ u_i=0,\quad i=1,\dots,k,&\text{on }\partial\Omega, \end{cases} \end{equation*} where $k\geq2$, $\Omega\subset\mathbb R^N$ ($N\geq4$) is a bounded domain with smooth boundary $\partial\Omega$, $2^*={2N}/({N-2})$ is the Sobolev critical exponent, $\mu_i\in\mathbb R$ for all $i=1,\dots,k$ are constants and $\lambda\in\mathbb R$ is a parameter. By the variational method, we mainly prove that the above system has a ground state for all $\lambda> 0$. Our results reveal some new properties of the above system that imply that the parameter $\lambda$ plays the same role as in the following well-known Brezis-Nirenberg equation \begin{equation*} \begin{cases} -\Delta u =\lambda u+ |u|^{2^*-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \end{equation*} and this system has a very similar structure of solutions as the above Brezis-Nirenberg equation for $\lambda$.

In this paper, we have used variational methods to study existence of solutions for the following critical nonlocal fractional Hardy elliptic equation \begin{equation*} (- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{|u|^{2_s^*(b) - 2} u}{|x|^{b}} + \lambda f (x, u ), \quad \text{in } \mathbb{R}^N, \end{equation*} where $N > 2 s $, $ 0< s< 1 $, $ \gamma, \lambda $ are real parameters, $(- \Delta)^s$ is the fractional Laplace operator, $2_s^*(b) = {2 (N - b)}/(N - 2s)$ is a critical Hardy-Sobolev exponent with $b \in [0, 2s)$ and $ f \in C(\mathbb R^{N} \times \mathbb{R}, \mathbb{R})$.

In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions is for a class of logistic type quasilinear problems in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine comparison principle with our result of existence of solution.

In this paper we provide a criterion for global secondary bifurcation via symmetry breaking. As an application, the occurrence of period-doubling bifurcations for the Lugiato-Lefever equation is proved.

We compute the Euler characteristic with compact supports $\chi_c$ of the formal barycenter spaces with weights of some locally compact spaces, connected or not. This reduces to the topological Euler characteristic $\chi$ when the weights of the singular points are less than one. As foresighted by Andrea Malchiodi, our formula is related to the Leray-Schauder degree for mean field equations on a compact Riemann surface obtained by C.C. Chen and C.S. Lin.

The classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.