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2002 Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations
Y. Du, Z. Guo
Adv. Differential Equations 7(12): 1479-1512 (2002).

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.

Citation

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Y. Du. Z. Guo. "Liouville type results and eventual flatness of positive solutions for $p$-Laplacian equations." Adv. Differential Equations 7 (12) 1479 - 1512, 2002.

Information

Published: 2002
First available in Project Euclid: 27 December 2012

zbMATH: 1075.35522
MathSciNet: MR1920542

Subjects:
Primary: 35J60
Secondary: 35B40

Rights: Copyright © 2002 Khayyam Publishing, Inc.

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Vol.7 • No. 12 • 2002
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