## Advances in Differential Equations

- Adv. Differential Equations
- Volume 7, Number 12 (2002), 1479-1512.

### 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.

#### Article information

**Source**

Adv. Differential Equations, Volume 7, Number 12 (2002), 1479-1512.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651584

**Mathematical Reviews number (MathSciNet)**

MR1920542

**Zentralblatt MATH identifier**

1075.35522

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Du, Y.; Guo, Z. Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations. Adv. Differential Equations 7 (2002), no. 12, 1479--1512. https://projecteuclid.org/euclid.ade/1356651584