## Differential and Integral Equations

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

### 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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 6 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367871527

**Mathematical Reviews number (MathSciNet)**

MR1392091

**Zentralblatt MATH identifier**

0855.35039

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

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