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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
##### Keywords:
limit at infinity; radial entire solution; zero