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.

Let Ω⊂ℝ^{$n$}be an arbitrary open set. We characterize the space$W$^{1,1}_{loc}(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space$W$^{1,1}(ℝ^{$n$}).

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