M. Cecchi, Z. Dosla, M. Marini

On the dynamics of the generalized Emden-Fowler equation

We present some recent results dealing with the qualitative behavior of solutions of the quasilinear second order differential equation % (*) \begin{equation} \big[a(t)\Phi_p(x^{^{\prime}})\big]^{^{\prime}}-b(t)\Phi_q(x)=0 \tag{$*$} \end{equation} where $a$, $b$ are positive continuous real functions, and $\Phi_{j}(u)=|u|^{j-2}u$, $j>1$. Following a classification of solutions, proposed in the linear case in [3], we divide all the solutions of $(*)$ with respect to their asymptotic behavior into two classes. The existence and uniqueness is considered: a topological approach is employed and a fixed point result for operators associated to boundary value problems on a half-line is used. In addition, some asymptotic estimates are presented and the convergence of solutions to zero as $t\to\infty$ is studied.