Coincidence sets associated with second-order ordinary differential equations of logistic type

Abstract

This paper concerns the formation of a coincidence set for the positive solution of the boundary-value problem $-{\varepsilon} (\phi_p(u_x))_x= \phi_q(u)f(c(x)-u)$ in $I=(-1,1)$ with $u(\pm 1)=0$, where ${\varepsilon}$ is a positive parameter, $\phi_p(s)=|s|^{p-2}s,\ 1 < q \le p < \infty$, $f(s) \sim \phi_{\theta+1}(s)\ (s \to 0)$, $0 < \theta < \infty$ and $c(x)$ is a positive smooth function satisfying $(\phi_p(c_x))_x=0$ in $I$. The positive solution $u_{\varepsilon}(x)$ converges to $c(x)$ uniformly on any compact subset of $I$ as ${\varepsilon} \to 0$. It is known that if $c(x)$ is constant and $\theta < p-1$, then the solution coincides with $c(x)$ somewhere in $I$ for sufficiently small ${\varepsilon}$ and the coincidence set $I_{\varepsilon}=\{x\in I : u_{\varepsilon}(x)=c(x)\}$ converges to $I$ as $|I \setminus I_{\varepsilon}| \sim {\varepsilon}^{1/p}\ ({\varepsilon} \to 0)$. It is proved in this paper that even if $c(x)$ is variable and $\theta < 1$, then $I_{\varepsilon}$ has a positive measure and converges to $I$ with order $O({\varepsilon}^{\kappa})$ as ${\varepsilon} \to 0$, where $\kappa=\min\{1/p,1/2\}$. Moreover, it is also shown that, if $\theta \ge 1$, then $I_{\varepsilon}$ is empty for every ${\varepsilon}$. The proofs rely on comparison principles and an energy method for obtaining local comparison functions.