Innovation based on Gaussian elimination to compute generalized inverse AT,S(2)

In this paper, we execute elementary row and column operations on the partitioned matrix $\left(\begin{array}{cc}\hfill GAG\hfill & \hfill G\hfill \\ \hfill G\hfill & \hfill 0\hfill \end{array}\right)$ into $\left(\begin{array}{cc}\hfill \left(\begin{array}{cc}\hfill I_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?A_{T,S}^{\left(2\right)}\hfill \end{array}\right)$to compute generalized inverse $A_{T,S}^{\left(2\right)}$ of a given complex matrix $A$, where $G$ is a matrix such that $R\left(G\right)=T$ and $N\left(G\right)=S$. The total number of multiplications and divisions operations is $T(m,n,s)=2m{n}^2+\frac{4m?s?1}{2}ns+(m?s)ns+mns$ and the upper bound of $T(m,n,s)$ is less than $6m{n}^2?\frac{3}{2}{n}^3?\frac{1}{2}{n}^2$ when $n\le m$. A numerical example is shown to illustrate that this method is correct.     

Existence of ground state solutions of Nehari–Pohozaev type for fractional Schrödinger–Poisson systems with a general potential

In this paper, we consider the following fractional Schrödinger–Poissonproblem

where $0<s\le t<1$ and $2s+2t>3$, the potential $V\left(x\right)$ is weakly differentiable and $f\in C(\mathbb{R},\mathbb{R})$. By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on $V$ and $f$. The results here extend the existing study.     

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems

where $\Omega$ is a bounded convex domain in ${\mathbb{R}}^n(n\ge 2)$, weighted nonlocal source satisfies $a\left(x\right)f\left(u(x,t)\right)\le a_1+a_2{\left(u(x,t)\right)}^p{\left({\int }_{\Omega }{\left(u(x,t)\right)}^l\mathrm{d}x\right)}^m,$ and $a_1,a_2,p,l,$ and $m$ are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.     

Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production

This paper deals with the following quasilinear chemotaxis-growth system

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n(n\ge 3)$ under zero-flux boundary conditions. The parameters $\mu ,\delta$ and $\tau$ are positive and the diffusion function $D\left(u\right)$ is supposed to generalize the prototype $D\left(u\right)\ge D_0{u}^{\theta }$ with $D_0>0$ and $\theta \in \mathbb{R}$. Under the assumption $\theta >1?\frac{4}{n}$, it is proved that whenever $\mu >0$, $\tau >0$ and $\delta >0$, for any given nonnegative and suitably smooth initial data ($u_0$, $v_0$, $w_0$) satisfying $u_0?0$, the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the $\left|\right|v(?,t)|{|}_{W^{1,s}\left(\Omega \right)}$ with $s\in [1,\frac{2n}{n?2})$ to estimate the boundedness of $\left|\right|?v(?,t)|{|}_{L^{2q}\left(\Omega \right)}(q>1)$. Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that $D\left(u\right)\equiv 1$ and $n=3$.     

Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega ?{\mathbb{R}}^2$, where the parameters $\chi ,\mu >0$ and $r\in \mathbb{R}$. Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.     

Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent

In this work, we are interested in considering the following nonlocal problem

where $\Omega ?{\mathbb{R}}^N(N\ge 4)$ is a smooth bounded domain, $a\ge 0,b>0,1<q<2,\mu ,\lambda >0$ and $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.     

Global bifurcation of positive radial solutions for an elliptic system in reactor dynamics

In this paper, we are concerned with the existence of positive radial solutions of the elliptic system $\{\begin{array}{c}?\Delta u=uv?\lambda u+f(\left|x\right|,u)\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ ?\Delta v=\mu u\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ u=v=0\text{,}\text{on }\left|x\right|=R_1\text{ and }\left|x\right|=R_2\text{,}\hfill \end{array}$ where $\left|x\right|={\left({\sum }_{i=1}^N{x}_i^2\right)}^{\frac{1}{2}}$, $\lambda >0$ is a constant, $\mu >0$ is a parameter and $0<R_1<R_2<\infty$, $f:[R_1,R_2]\times [0,\infty )\to [0,\infty )$ is continuous and $f(t,s)>0$ for all $(t,s)\in [R_1,R_2]\times (0,\infty )$. Under some appropriate conditions on the nonlinearity $f$, we show that the above system possesses at least one positive radial solution for any $\mu \in (0,\infty )$. The proof of our main results is based upon bifurcation techniques.     

Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential

In this work, we are interested in studying the following Kirchhoff type problem

where $\Omega ?{\mathbb{R}}^N(N\ge 3)$ is a smooth bounded domain, $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent, $0<q<1,\lambda >0$, and $f\in L^{\infty }\left(\Omega \right)$ with the set $\{x\in \Omega :f\left(x\right)>0\}$ of positive measures, and $g\in L^{\infty }\left(\Omega \right)$ with $g\left(x\right)\ge 0,g?0.$ By the Nehari method and variational method, the existence of positive ground state solutions is obtained.