Let $\Omega\subset\mathbb{R}^{N}$, $N\geq1$, be a smooth bounded domain, and let $m:\Omega\rightarrow\mathbb{R}$ be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem \[ \left\{ \begin{array} [c]{lll} -\Delta u=\lambda m(x)(f(u)-k) & \mathrm{in} & \Omega,\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where $\lambda,k>0$ and $f$ is either sublinear at infinity with $f(0)=0$, or $f$ has a singularity at $0$. We prove the existence of a positive solution for certain ranges of $\lambda$ provided that the negative part of $m$ is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.

We show that the cubic Dirac equation, also known as the Thirring model, scatters at infinity to a linear solution modulo a phase correction.

The present paper deals with uniform stability for non-autonomous impulsive systems. We consider a non-autonomous system with impulses in its abstract form and we present conditions to obtain uniform stability, uniform asymptotic stability and global uniform asymptotic stability using Lyapunov functions. Using the results from the abstract theory we present sufficient conditions for a controlled predator-prey model under impulse conditions to be globally uniformly asymptotically stable.

This paper deals with a two-competing-species chemotaxis system with two different chemicals \begin{equation*} \begin{cases} u_{t}=d_{1}\Delta u-\chi_{1}\nabla \cdot(u\nabla v)+\mu_{1} u(1-u-a_{1}w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{2}\Delta v-\alpha_{1}v+\beta_{1}w, & (x,t)\in \Omega\times (0,\infty),\\ w_{t}=d_{3}\Delta w-\chi_{2}\nabla \cdot(w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \Omega\times (0,\infty), \\ 0=d_{4}\Delta z-\alpha_{2}z+\beta_{2}u, & (x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$ with nonnegative initial data $(u_{0},w_{0})\in (C^{0}(\overline{\Omega}))^{2}$ satisfying $u_{0}\not\equiv0$ and $w_{0}\not\equiv 0$, where $\chi_{1},\chi_{2}\geq0$, $a_{1}, a_{2}\in[0,1)$, and the parameters $d_{i}$ ($i=1,2,3,4$) and $\alpha_{j},\beta_{j}, \mu_{j}$ ($j=1,2$) are positive. Based on the approach of eventual comparison, it is shown that under suitable conditions, the system possesses a unique global-in-time classical solution, which converges to the constant steady states.

The aim of the present paper is to contribute to the development of the study of Cauchy problems involving Riemann-Liouville and Caputo fractional derivatives. First, existence-uniqueness results for solutions of non-linear Cauchy problems with vector fractional multi-order are addressed. A qualitative result about the behavior of local but non-global solutions is also provided. Finally, the major aim of this paper is to introduce notions of fractional state-transition matrices and to derive fractional versions of the classical Duhamel formula. We also prove duality theorems relying left state-transition matrices with right state-transition matrices.

The aim of this paper is to describe the asymptotic behavior, as $\varepsilon\to 0$, of an elliptic problem with rapidly oscillating coefficients in an $\varepsilon$-periodic two component composite with an interfacial contact resistance on the interface, in the case of weakly converging data.

Using global diffeomorphism theorem based on duality mapping and mountain geometry, we investigate the properties of the Volterra operator given pointwise for $t\in \left[ 0,1\right] $ by \begin{equation*} V(x)(t)=x(t)+ \int _{0}^{t} v(t,\tau ,x(\tau ))d\tau ,\text{ }x(0)=0. \end{equation*}

We study the existence of positive radial solutions to the problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u &= \lambda K_1(|x|) f(v) \hspace{.3in}\mbox{in } \Omega_e,\\ -\Delta_p v &= \lambda K_2(|x|) g(u) \hspace{.31in}\mbox{in } \Omega_e, \\u &= v=0 \hspace{.7in} \mbox{ if } |x|=r_0, \\u(x)&\rightarrow 0,v(x)\rightarrow 0 \hspace{.4in} \mbox{as }\left|x \right|\rightarrow\infty, \end{aligned} \right. \end{equation*} where $\Delta_p w:=\mbox{div}(|\nabla w|^{p-2}\nabla w)$, $1 < p < n$, $\lambda$ is a positive parameter, $r_0>0$ and $\Omega_e:=\{x\in\mathbb{R}^n|~|x|>r_0\}$. Here, $K_i:[r_0,\infty)\rightarrow (0,\infty)$, $i=1,2$ are continuous functions such that $\lim_{r \rightarrow \infty} K_i(r)=0$, and $f, g:[0,\infty)\rightarrow \mathbb{R}$ are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of $\lambda$ via degree theory and rescaling arguments.