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 $\mathbb{R}^n$

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.