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On the oscillation of solutions of elliptic equations. (English. Russian original) Zbl 0615.35026

Sov. Math., Dokl. 31, 82-85 (1985); translation from Dokl. Akad. Nauk SSSR 280, 300-303 (1985).
The author considers the equation \((-\Delta)^ m u+a(x)u=0\), \(a\in L_{\infty,loc}({\mathbb{R}}^ n)\), and calls this equation oscillatory if for each \(r_ 0>0\) there is some bounded domain G in the complement of the ball \(| x| \leq r_ 0\) such that the equation has a nontrivial solution in the Sobolev space \(\overset\circ W^ m_ 2(G)\). In the paper various conditions on a(x) – in the form of pointwise or integral inequalities – are given which imply oscillatory and nonoscillatory behaviour, respectively. The proofs consist in an estimation of the quadratic form associated with the equation.
Reviewer: F.Tomi

MSC:

35J30 Higher-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
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