Classification of positive solutions of $p$-Laplace equation with a growth term

Matteo Franca

 Dipartimento di Matematica, Universita' di Firenze, Viale Morgagni 67a, 50134 Firenze, Italy


We give a structure result for the positive radial solutions of the following equation: \begin{equation*} \Delta_{p}u+K(r) u|u|^{q-1}=0 \end{equation*} with some monotonicity assumptions on the positive function $K(r)$. Here $r=|x|$, $x \in \RR^n$; we consider the case when $n>p>1$, and $q >p_* =\frac{n(p-1)}{n-p}$. We continue the discussion started by Kawano et al. in \cite{KYY}, refining the estimates on the asymptotic behavior of Ground States with slow decay and we state the existence of S.G.S., giving also for them estimates on the asymptotic behavior, both as $r \to 0$ and as $r \to \infty$. We make use of a Emden-Fowler transform which allow us to give a geometrical interpretation to the functions used in \cite{KYY} and related to the Pohozaev identity. Moreover we manage to use techniques taken from dynamical systems theory, in particular the ones developed in \cite{JPY2} for the problems obtained by substituting the ordinary Laplacian $\Delta$ for the $p$-Laplacian $\Delta_{p}$ in the preceding equations.

AMSclassification. 37D10, 35H30.

Keywords.  $p$-Laplace equations, radial solution, regular/singular ground state, Fowler inversion, invariant manifold.