Existence of nodal fast-decay solutions to \(\text{div}(|\nabla u|^{m-2}\nabla u) + K(| x|)| u|^{q-1}u = 0\) in \(\mathbb{R}^ n\). (English) Zbl 0855.35039

Kabeya, Yoshitsugu; Yanagida, Eiji; Yotsutani, Shoji

Differ. Integral Equ. 9, No. 5, 981-1004 (1996).

The authors study the quasilinear elliptic equation \[ \text{div}(|\nabla u|^{m- 2} \nabla u)+ K(|x|) |u|^{q- 1} u= 0,\;x\in \mathbb{R}^n,\;\lim_{|x|\to \infty} |x|^{{n- m\over m- 1}} |u(|x|)|> 0, \] where \(1< m< n\), \(q> m- 1\), \(K(r)\in C^1((0, \infty))\) and \(K(r)> 0\) on \((0, \infty)\). They 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, and Y. Naito for \(m= 2\), but the proofs are considerably different from their even if \(m= 2\). Finally, they consider various further boundary problems with similar results.

Reviewer: Michael Sever (Jerusalem)

MSC:

35J60	Nonlinear elliptic equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

radial solutions; prescribed numbers of zeros