In this paper, based on the parabolic interpolation inequality and inductive argument, we study the space-time decay estimates of higher-order time and spatial derivatives of strong solutions for the 3D incompressible viscous Camassa-Holm equations provided that the initial datum is well localized.

We show that ground state solutions to the nonlinear, fractional problem \begin{equation*} \begin{cases} (-\Delta)^{s} u + V(x) u = f(x,u) & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb R^N \setminus \Omega, \end{cases} \end{equation*} on a bounded domain $\Omega \subset \mathbb R^N$, converge (along a subsequence) in $L^2 (\Omega)$, under suitable conditions on $f$ and $V$, to a solution of the local problem as $s \to 1^-$.

We study the following Schrödinger-Poisson-Slater type system: \begin{equation}\tag{1} \begin{cases} -\Delta u + V(x) u + \lambda\phi (x) u =|u|^{p-1}u,& x\in \mathbb{R}^3, \\ -\Delta\phi = u^2, \quad \lim\limits_{|x|\to +\infty}\phi (x)=0, \end{cases} \end{equation} where $\lambda> 0$ is a parameter, $p\in (1,2)$, $V\in C(\mathbb{R}^N,\mathbb{R}_0^+)$. If $V$ is a constant, or $p \in (2,5)$ with $V\in L^{\infty}(\mathbb{R}^3),$ the above system has been considered in many papers. In this paper, we are interested in the case: $p\in (1,2)$ and $V$ being a coercive potential, i.e., $\lim\limits_{|x|\to +\infty} V(x)=\infty$. We prove that there exists $\lambda_0> 0$ such that system (1) has at least two positive solutions $u^0_\lambda$ and $u^1_\lambda$ for any $\lambda \in (0, \lambda_0)$. Moreover, $u^0_\lambda$ is a ground state (i.e., the least energy solution) which must blow up as $\lambda \rightarrow 0$. Particularly, when $p\in ({11}/{7},2)$ and $\lambda> 0$ is small enough, we show that the ground state of (1) must be non-radially symmetric even if $V(x)=V(|x|)$, such as $V(x)=|x|^2$.

In this paper we characterize the phase portraits in the Poincaré disc of the class of polynomial differential systems of the form \begin{equation*} \dot{x}=-y,\qquad \dot{y} =x+ax^{4}+bx^{2}y^{2}+cy^{4}, \end{equation*} with $a^2+b^2+c^2\neq0$, which are symmetric with respect to the $x$-axis. Such systems have a center at the origin of coordinates. Moreover, using the averaging theory of five order, we study the number of limit cycles which can bifurcate from the period annulus of this center when it is perturbed inside the class of all polynomial differential systems of degree $4$.

We study the initial-boundary value problem for the so-called alpha model of Jeffreys-Oldroyd fluids motion in 2D and 3D dimensions. In this paper the global in time existence of dissipative solutions to this problem is obtained. For this the topological approximation method to the initial-boundary value problem is applied.

We study the Cauchy problem for nonlinear Schrödinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and orbital stability of standing waves for the problem in the mass-subcritical and mass-critical cases. In the mass-critical case, we give a detailed description of the blow-up behavior of standing waves when the mass tends to a critical value.

The concept of the homotopic distance and its higher analogs are introduced in [7]. In this paper we introduce some important properties of higher homotopic distances, investigate the conditions under which cat, secat and higher dimensional topological complexity categories are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distances, to some TC$_n$-related theorems.

We are concerned with Dirichlet problems of the form $$ {\rm div}(|D u|^{p-2}Du)+f(u)=0\quad \mbox{in }\Omega,\qquad u=0\quad \mbox{on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1< p< n$ and $f$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $n=2$ and $\gamma\colon [a,b]\to\mathbb{R}^2$ is a smooth curve such that $\gamma(t_1)\neq\gamma(t_2)$ for $t_1\neq t_2$, we prove that, for $\varepsilon> 0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $\Omega=\Omega^\Gamma_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2 : {\rm dist}\big((x_1,x_2),\Gamma\big)< \varepsilon\}$, where $\Gamma=\{\gamma(t) : t\in[a,b]\}$. Moreover, we extend this uniqueness result to the case where $n> 2$ and $\Omega$ is, for example, a domain of the type $$ \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y) : (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|< s\}. $$

Motivated by classical Euler's Tonnetz, we introduce and study the combinatorics and topology of more general simplicial complexes Tonn$^{n,k}(L)$ of Tonnetz type. Out main result is that for a sufficiently generic choice of parameters the generalized Tonnetz Tonn$^{n,k}(L)$ is a triangulation of a $(k-1)$-dimensional torus $T^{k-1}$. In the proof we construct and use the properties of a discrete Abel-Jacobi map, which takes values in the torus $T^{k-1} \cong \mathbb{R}^{k-1}/\Lambda$ where $\Lambda \cong \mathbb{A}^\ast_{k-1}$ is the permutohedral lattice.

In this work, we study the existence and nonexistence of positive radial solutions for the quasilinear equation $\mathrm{div}(A(|\nabla u|)\nabla u)+\lambda k(|x|)f(u)=0$ in the exterior of a ball with vanishing boundary conditions using an approach based on a fixed point theorem for operators on Banach Space.

In the paper, we consider the following boundary value problem involving the fractional $p$-Laplacian: \begin{equation} \tag{$\mathcal{P}$} \begin{cases} (-\triangle)_p^su(x)=f(x,u) &\text{in } \Omega,\\ u(x)=0 &\text{in } \mathbb{R}^N\setminus\Omega. \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 1$, $(-\Delta)_p^{s}$ is the fractional $p$-Laplacian with $s\in (0,1)$, $p\in(1,{N}/{s})$, $f(x, u)\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$. Under the improved subcritical polynomial growth condition and other conditions, the existences of a least-energy sign-changing solution for the problem $(\mathcal{P})$ has been established.

We show that the typical nonexpansive mapping on a small enough subset of a ${\rm CAT}(\kappa)$-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $\sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.

In this paper we study the existence of nontrivial solutions for the fractional Laplacian resonance problem with a bounded nonlinearity via Morse theory and a penalization technique.

We consider a quasilinear Robin problem driven by the $p$-Laplacian plus a positive potential and with a small perturbation. We assume that the main term in the equation has an Ekeland structure but we do not suppose any growth condition for the perturbation term. Applying variational methods, we prove the existence of at least one nontrivial weak solution.

We show that the regularity set of any family of sequences of matrices is an $F_{\sigma\delta}$-set. Moreover, for any $F_{\sigma\delta}$-set containing zero, we construct a family of sequences of matrices having this regularity set.

In this paper, we study the second-order boundary asymptotic behaviour for large positive solutions to an elliptic system of competitive type. First, we derive a second-order estimate to a related single weighted equation with boundary blow-up data. Then, by relaxing the system and iterating the estimate of the single equation, we establish second-order estimates of the solutions.