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The equation$x$^{(n)}(t)=(−1)^{$n$}│$x(t)$│^{$k$}with$k$>1 is considered. In the case$n$≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with$C$(t−t_{0})^{−n/(k−1)}, where$C$is a constant depending only on$n$and$k$. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times ($t−t$_{0})^{−n/(k−1)}. It is shown that they do not necessarily coincide with$C$(t−t_{0})^{−n/(k−1)}. This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.

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No abstract uploaded!

We study the asymptotic behaviour of$N$(α)—the number of negative eigenvalues of the operator (-τ)^{$l$}-α$V$in$L$_{2}($R$^{$d$}) for an even$d$and$2l≥d$. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases where$N$(α) is of the Weyl-type,$N$(α) is of the Weyl-order but not the Weyl-type coefficient and finally where$N$(α)=$O$(α^{q}) with$q>d/2l$.