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Zbl 0615.35026
Toraev, A.
On the oscillation of solutions of elliptic equations. (English. Russian original)
[J] Sov. Math., Dokl. 31, 82-85 (1985); translation from Dokl. Akad. Nauk SSSR 280, 300-303 (1985). ISSN 0197-6788

The author considers the equation $(-\Delta)\sp m u+a(x)u=0$, $a\in L\sb{\infty,loc}({\bbfR}\sp n)$, and calls this equation oscillatory if for each $r\sb 0>0$ there is some bounded domain G in the complement of the ball $\vert x\vert \le r\sb 0$ such that the equation has a nontrivial solution in the Sobolev space $\overset\circ\to W\sp m\sb 2(G)$. 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.
[F.Tomi]

MSC 2000:: *35J30 Higher order elliptic equations, general
35B05 General behavior of solutions of PDE
35J40 Higher order elliptic equations, boundary value problems

Keywords: oscillatory; Sobolev space; inequalities; nonoscillatory

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