## Asymptotic properties of ground states of scalar field equations with a vanishing parameter    [PDF]

Vitaly Moroz, Cyrill B. Muratov
We study the leading order behaviour of positive solutions of the equation
-\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete
characterization of all possible asymptotic regimes as a function of $p$, $q$
and $N$. The behavior of solutions depends sensitively on whether $p$ is less,
equal or bigger than the critical Sobolev exponent $p^\ast=\frac{2N}{N-2}$. For
$pequation in which the last term is absent. For$p>p^\ast$the solution asymptotically coincides with the solution of the equation with$\varepsilon=0$. In the most delicate case$p=p^\ast$the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden--Fowler equation, whose choice depends on$\varepsilon\$ in a nontrivial
way.
View original: http://arxiv.org/abs/1202.3426