Differential and Integral Equations

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, and Shoji Yotsutani

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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.

Article information

Differential Integral Equations, Volume 9, Number 5 (1996), 981-1004.

First available in Project Euclid: 6 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Kabeya, Yoshitsugu; Yanagida, Eiji; Yotsutani, Shoji. 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 (1996), no. 5, 981--1004. https://projecteuclid.org/euclid.die/1367871527

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