Classification of proper solutions to some Emden-Fowler equations
The author develops an asymptotic theory on the second-order Emden-Fowler equation \[ u''\pm (1+\varepsilon(t)) e^{bt}|u|^\lambda \text{sgn }u= 0,\quad t\geq t_0> 0,\tag{1} \] and he states analogous results to those obtained by Bellman, Chanturiya and others. Equation (1) is classified into four types according to the signs of the coefficient function and the exponent \(b\), namely: \[ u''+ (1+\varepsilon(t)) e^{\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{++}\)} \] \[ u''+ (1+\varepsilon(t)) e^{-\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{+-}\)} \] \[ u''- (1+ \varepsilon(t)) e^{\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{-+}\)} \] \[ u''- (1+\varepsilon(t)) e^{-\alpha t}|u|^\lambda \text{sgn }u= 0.\tag{E\(_{--}\)} \] The main goal is to determine the asymptotic forms of positive proper solutions to \((\text{E}_{+-})\), \((\text{E}_{-+})\), \((\text{E}_{--})\) and also to classify these solutions. To start with we mention that a nontrivial solution \(u\) to (1) is said to be proper if it is defined in some neighborhood of \(+\infty\), and is nontrivial in any neighborhood of \(+\infty\). Firstly, it is shown that if \(\lambda\in \mathbb{R}\setminus \{0,1\}\), \(\alpha> 0\) and \(\lim_{t\to\infty} \varepsilon(t)= 0\), then all positive proper solutions \(u\) to \[ u''+ (1+\varepsilon(t)) e^{-\alpha t} u^\lambda= 0\tag{E\(_{+-}\)} \] possess one of the following asymptotic forms \[ u= a_1t+ a_2- {(a_1t)^2\over \alpha^2} e^{-\alpha t} (1+ O(1)); \] \[ u= a-{a^\lambda\over \alpha^2} e^{-\alpha t} (1+ O(1)), \] where \(a>0\), \(a_1> 0\), \(a_2\) are constants. If \(\lambda> 1\), \(\varepsilon(t)\to 0\) as \(t\to\infty\) or if \(0< \lambda< 1\) and either \[ \lim_{t\to\infty} \varepsilon(t)= 0,\quad \int^\infty|\varepsilon'(t)|dt< \infty \] or \[ \lim_{t\to\infty} \varepsilon(t)= 0,\quad \int^\infty|\varepsilon(t)|^2 dt< \infty \] hold, then all positive proper solutions \(u\) to \[ u''- (1+ \varepsilon(t)) e^{\alpha t} u^\lambda= 0\tag{E\(_{-+}\)} \] possess the asymptotic from \[ u\sim u_0\equiv ce^{\beta t},\quad \beta=-{\alpha\over\lambda- 1},\quad c=|\beta|^{2/(\lambda- 1)}. \] In resembling conditions as in the case of the \((\text{E}_{-+})\) equation it is proved that all positive proper solutions to \[ u''- (1+ \varepsilon(t)) e^{-\alpha t} u^\lambda= 0\tag{E\(_{--}\)} \] possess one of the following asymptotic forms: \[ u= a_1t+ a_2+ {(a_1t)^\lambda\over \alpha^2} e^{-\alpha t}(1+ O(1)); \] \[ u= a+ {a^\lambda\over \alpha^2} e^{-\alpha t}(1+ O(1)); \] \[ u\sim u_0\equiv ce^{\beta t},\quad \beta= {\alpha\over\lambda- 1},\quad c=|\beta|^{2/(\lambda- 1)}. \] Moreover, in appropriate conditions the author states the uniqueness of positive decaying proper solutions to \((\text{E}_{--})\), i.e. the solution satisfying \(\lim_{t\to\infty} u(t)= 0\).
scientific article; zbMATH DE number 1423814