Open Access
Translator Disclaimer
1996 Existence of nodal fast-decay solutions to ${\rm div}(\vert \nabla u\vert ^{m-2}\nabla u)+K(\vert x\vert )\vert u\vert ^{q-1}u=0$ in $\mathbb{R}^n$
Yoshitsugu Kabeya, Eiji Yanagida, Shoji Yotsutani
Differential Integral Equations 9(5): 981-1004 (1996).

Abstract

We study the quasilinear elliptic equation $$ \left\{\eqalign{&\text{div}|\nabla u|^{m-2}+K(|x|)|u|^{q-1}=0,\quad x\in\mathbb{R}^n,\cr&\lim_{|x|\to \infty}|x|^{{n-m\over m-1}} | u(|x|) | >0,\cr}\right. $$ where $1<m<n,$ $ q>m-1,$ $ K(r)\in C^{1}((0,\infty))$ and $ K(r)>0 $ on $ (0,\infty).$ We show the existence of radial solutions with prescribed numbers of zeros under simple conditions. These results are generalizations of those due to E\. Yanagida and S\. Yotsutani ([17]) and Y. Naito ([13]) for $m=2$, but the proofs are considerably different from theirs even if $m=2$. Finally, we consider various further boundary problems with similar results.

Citation

Download Citation

Yoshitsugu Kabeya. Eiji Yanagida. Shoji Yotsutani. "Existence of nodal fast-decay solutions to ${\rm div}(\vert \nabla u\vert ^{m-2}\nabla u)+K(\vert x\vert )\vert u\vert ^{q-1}u=0$ in $\mathbb{R}^n$." Differential Integral Equations 9 (5) 981 - 1004, 1996.

Information

Published: 1996
First available in Project Euclid: 6 May 2013

zbMATH: 0855.35039
MathSciNet: MR1392091

Subjects:
Primary: 35J60
Secondary: 35B05

Rights: Copyright © 1996 Khayyam Publishing, Inc.

JOURNAL ARTICLE
24 PAGES


SHARE
Vol.9 • No. 5 • 1996
Back to Top