Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations

Abstract

For a wide class of nonlinearities $f(u)$ satisfying \[\mbox{ $f(0)=f(a)=0$, $f(u)>0$ in $(0,a)$ and $f(u) < 0$ in $(a,\infty)$,}\] we study the quasilinear equation $-\Delta_p u=\lambda c(x)f(u)$ over the entire $\mathbb{R}^N$ or over a bounded smooth domain $\Omega$. Such equation covers various models from chemical reaction theory and population biology. We show that any nonnegative solution on the entire $\mathbb{R}^N$ must be a constant, and for large $\lambda$, any positive solution on $\Omega$ must approach $a$ in compact subsets of $\Omega$, no matter whether or not the solution has a prescribed behavior near the boundary of $\Omega$. We also determine the flat cores of the positive solutions and show that the flat cores enlarge to the whole $\Omega$ as $\lambda$ goes to infinity. Our proof of these results demonstrates the usefulness of boundary blow-up solutions in various classical problems.