zbMATH — the first resource for mathematics

Existence and asymptotic behavior of positive solutions of second order quasilinear differential equations. (English) Zbl 1025.34043
Here, the existence and the asymptotic behavior of positive solutions to the equation \[ (p(t)|y'|^{\alpha -1}y')'=q(t)|y|^{\beta -1}y ,\quad t\geq a\geq 0 , \] where \(\alpha, \beta\) are positive constants, \(p(t), q(t)\) are positive continuous functions defined on \([a,\infty)\), and \(\int_a^{\infty}(p(t))^{(-1/\alpha)} dt <\infty\). Criteria for existence and nonexistence of singular positive solutions are obtained. It is proved that there are six types of different asymptotic behavior as \(t\to\infty\) for proper positive solutions. Conditions for the existence of each of these types are given. The obtained results for ordinary equations are applied to investigate radial symmetric solutions to the PDE \[ \text{div}(|Du|^{m-2}Du)=c(|x|)|u|^{n-2}u , \] with \(m, n > 1\), \(s=(x_1,\ldots,x_N)\), \(Du=(\partial u/\partial x_1,\ldots,\partial u/\partial x_N)\).

34D05 Asymptotic properties of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems