Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=\Delta u-\chi_1\nabla(u\nabla v_1)+\chi_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=\Delta v_1-\lambda_1v_1+\mu_1u,\ x\in\mathbb{R}^N,\\ 0=\Delta v_2-\lambda_2v_2+\mu_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+\chi_2\mu_2>\chi_1\mu_1+K$, then the steady solution $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+\chi_2\mu_2>\chi_1\mu_1+K$, then there exist a number $c^*(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\geq 2\sqrt a$ such that for every $c\in ( c^*(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) , \infty)$ and $\xi\in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdot\xi-ct),V_1(x\cdot\xi-ct),V_2(x\cdot\xi-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ and $(0,0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover we prove that$$\lim_{(\chi_1,\chi_2)\to(0^+,0^+)}c^{*}(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=\begin{cases}2\sqrt a\ \text{if}\ a\leq \min\{\lambda_1, \lambda_2\}\\ \frac{a+\lambda_1}{\sqrt{\lambda_1}}\ \text{if}\ \lambda_1\leq \min\{a, \lambda_2\}\\ \frac{a+\lambda_2}{\sqrt{\lambda_2}}\ \text{if}\ \lambda_2\leq \min\{a, \lambda_1\}\end{cases},\forall \lambda_1,\lambda_2,\mu_1,\mu_2>0,$$and$$\lim_{x\to\infty}\frac{U(x)}{e^{-\sqrt{a}\mu x}}=1,$$ where $\mu$ solves $\sqrt a(\mu+\frac{1}{\mu})=c$ in the interval $(0 , \min\{1,\sqrt{\frac{\lambda_1}{a}},\sqrt{\frac{\lambda_2}{a}}\})$.