Positive steady states in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1)$\left(\begin{array}{cc}-d_1\Delta u=\lambda u-u^2-buv,& x\in \Omega ,\\ -\text{div}[d\left(w\right)\nabla v+\chi \left(w\right)v\nabla w]=\mu v-v^2-cuv,& x\in \Omega ,\\ -d_3\Delta w=-\kappa w+\tau u,& x\in \Omega ,\\ u=v=w=0,& x\in \partial \Omega \end{array}\right)$in a bounded smooth domain $\Omega \subset R^N(N\ge 1)$, where $\lambda ,\mu ,b,c,d_1,d_3,\kappa ,\tau$ are positive constants, and $(d\left(w\right),\chi \left(w\right))\in \text{[}{C}^1{[0,\infty \text{)}]}^2$ with $d\left(w\right),\chi \left(w\right)>0$ for all $w\ge 0$. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with $d\left(w\right),\alpha \left(w\right)$ being arbitrary functions in $w$, this makes the analysis of system (0.1) much more difficult. By the $W^{2,p}$-estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form $(u,0,w)$ with $u,w>0$ and the semitrivial solution of the form $(0,v,0)$ with $v>0$, which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states.