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Radial entire solutions of the linear equation \(\Delta u+\lambda p(| x|)u=0\). (English) Zbl 0716.35002
The author studies radial entire solutions of the linear elliptic equation \(\Delta u+\lambda p(| x|)u=0\), \(x\in {\mathbb{R}}^ N\), where \(N\geq 3\) and the function p satisfies
p\(\in C[0,\infty)\), \(p\geq 0\) on \([0,\infty)\), and \(p\not\equiv 0\) on \([T,\infty)\) for all \(T\geq 0.\)
It is obtained the following theorem:
(I) Assume that \(\int^{\infty}_{0}t p(t)dt<\infty\). There exist \(\lambda_ 0\) and \(\lambda_ 1\) with \(0<\lambda_ 0\leq \lambda_ 1<\infty\) such that
(i) if \(\lambda \in (0,\lambda_ 0)\), then every nontrivial radial entire solution u(t) has no zero in \([0,\infty)\) and \(\lim_{t\to \infty}u(t)\) exists and is a non-zero finite value;
(ii) if \(\lambda \in [\lambda_ 0,\lambda_ 1]\), then every nontrivial radial entire solution u(t) has no zero in \([0,\infty)\) and \(\lim_{t\to \infty}t^{N-2}u(t)\) exists and is a non-zero finite value;
(iii) if \(\lambda \in (\lambda_ 1,\infty)\), then every nontrivial radial entire solution has at least a zero in \([0,\infty).\)
(II) Assume that \(\int^{\infty}_{0}t^{N-1}p(t)dt<\infty\). Then, in addition to \(\lambda_ 1\) in (I), there exist \(\lambda_ k\) \((k=2,3,...)\) with \(0<\lambda_ 1<\lambda_ 2<...<\lambda_ k<\lambda_{k+1}<..\). and \(\lim_{k\to \infty}\lambda_ k=\infty\) such that if \(\lambda \in (\lambda_ k,\lambda_{k+1}]\) \((k=1,2,...)\), then every nontrivial radial entire solution has exactly k zeros in \([0,\infty).\)
As the author points out, there is a rather extensive literature on entire solutions for semilinear equations similar to the above linear equation, but the linear case requires a specific study.
Reviewer: F.Bernis

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
35J15 Second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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