517.925We establish necessary and sufficient conditions for the existence of one class of solutions of a differential equation with exponential nonlinearity. Asymptotic representations of these solutions are obtained.
Formulation of Main Theorems and Auxiliary StatementsConsider the differential equationFor n = 2, the asymptotic behavior of solutions of Eq. (1) was investigated by Evtukhov and Drik in [1-5]. In [6], for arbitrary n, existence conditions and asymptotic representations were obtained for solutions of Eq. (1) with property P i ω , i ∈ {0, . . . , n − 1}, i.e., for solutions that satisfy the conditions lim t↑ω y (i) (t) = c i = 0, lim t↑ω y (k) (t) = either ±∞, or 0, k = i + 1, . . . , n − 1. A solution y of Eq. (1) is called aP ω (λ 0 n−1 )-solution if it is defined on a certain interval [t 0 , ω[⊂ [a, ω[ and the function z(t) = e y(t) satisfies the following conditions: (i) z (n) (t) = 0 for t ∈ [t 0 , ω[ ; (ii) for any k ∈ {0, 1, . . . , n − 1}, one has either lim t↑ω z (k) (t) = 0 or lim t↑ω z (k) (t) = ±∞ ;(iii) the following limit exists and is either finite or equal to ±∞:furthermore, if λ 0 n−1 = 1, then, for any k ∈ {1, . . . , n − 1}, one has either lim t↑ω z (t) z(t) (k−1)= 0
