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Investigation of the oscillatory properties of solutions to partial differential equations using sphere means. (English. Russian original) Zbl 0967.35036
Dokl. Math. 57, No. 3, 413-415 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 360, No. 4, 448-450 (1998).
From the introduction: In the investigation of the oscillation properties of solutions to elliptical partial differential equations with the Laplacian in the principal term, a sphere mean of the form $M_r[u(x), P_0]= {1\over \omega_n r^{n-1}} \int\underset{S_r}\cdots\int u(x) ds$ has been widely used since the middle 1970s. Here, we use this approach to investigate the oscillation properties of partial differential equations in spaces $$X$$ of constant curvature of the form $L^nu+ p(x)u= 0,$ where $$L$$ is the Laplace-Beltrami operator having different form for each particular space $$X$$ of constant curvature.
##### MSC:
 35J15 Second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs