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

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*Matteo Franca*

**Address.**

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

**E-mail. **franca@math.unifi.it

**Abstract.**

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.