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Radial solutions for \(\Delta u+| x| ^{\ell}| u| ^{p- 1}=0\) on the unit ball in \(R^ n\). (English) Zbl 0699.35097

The nonlinear Dirichlet problem (P) \(\Delta u+| x|^{\ell}| u|^{p-1}u=0\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) is studied, where \(\Omega\) is the unit ball \(\{\) x \(|\) \(| x| <1\}\) in \({\mathbb{R}}^ n\) with \(n\geq 3\). Through the phase plane analysis the following results are proved: If \(p\in (1,(n+2+2\ell)/(n-2))\), for each positive integer k the problem (P) has a unique radial solution \(u=u(r)\) such that u(0) is positive and u(r) has exactly (k-1) zeros in (0,1). If \(p\in [(n+2+2\ell)/(n-2),\infty)\), (P) has no radial solution except for the trivial one. Further it is proved from Rellich’s identity that there exists no nontrivial solution of (P) in the latter case.
Reviewer: K.Nagasaki

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
34C99 Qualitative theory for ordinary differential equations
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