Optimal uniform regularity and asymptotic behavior of solutions to Lotka-Volterra type systems with strong competition and asymmetric coefficients

Abstract

In this paper, we investigate the uniform regularity and asymptotic behavior of solutions to the following Lotka-Volterra type system of strong competition with Dirichlet boundary conditions: \begin{align*} \left\{ \begin{array}{ll} -\Delta u_{i,\beta} = f_{i,\beta}(x, u_{i,\beta}) - \beta u_{i,\beta}^{p_i} \sum_{\substack{j=1 \\ j \neq i}}^k a_{ij} u_{j,\beta}^{p_j}, \quad u_{i,\beta} > 0 & \text{in } \Omega, u_{i,\beta} = \varphi_{i,\beta} & \text{on } \partial\Omega, \end{array}\right. \end{align*} where $N \geq 1$, $1 \leq i \leq k$ with $k \geq 3$, $\beta > 0$, $p_i \geq 1$, $a_{ij} > 0$ for $i \neq j$, and $\Omega$ is a $C^{1,\text{Dini}}$ bounded domain in $\mathbb{R}^N$. First, we prove that the uniform boundedness of the solutions implies their uniform interior and global Lipschitz boundedness as $\beta \to +\infty$. Such uniform results are optimal; partial versions thereof are known in the literature for symmetric coefficients (i.e., $a_{ij} = a_{ji}$ for all $i \neq j$) and homogeneous competition terms (i.e., $p_i = p_j$ for all $i\neq j$). Here, we establish an Alt-Caffarelli-Friedman type monotonicity formula for the system and then employ blow-up analysis to show that these results also hold in the asymmetric or nonhomogeneous case. Next, as consequences of the uniform optimal regularity, we derive sharp quantitative pointwise estimates for the densities near the interface between different components.