Seminario di analisi matematica
ore
15:30
presso Seminario II
In this talk we illustrate results concerning radial positive solutions of semi-linear elliptic equations such as
$$\Delta u + k(|x|) u^{q-1}=0 \qquad \qquad \qquad (1)$$
where $x \in \mathbb{R}^n$, $n>2$, $k(|x|)>0$, and its generalization to the $p$-Laplace case.
We focus in particular on the critical case $q=2^*=\frac{2n}{n-2}$. Our goal is to find conditions on $k$ ensuring existence and multiplicity
of ground states with fast decay, i.e. solutions $u(x)$ defined and positive in the whole of $\mathbb{R}^n$ and decaying as $|x|^{-(n-2)}$ for $|x|$ large.
Using Fowler transformation we pass from (1) to a two dimensional dynamical system so that we can apply
phase plane techniques such as invariant manifold theory, shooting, Melnikov theory. In particular
the search of ground states with fast decay is translated on the search of homoclinic trajectories.