We study the leading order behaviour of positive solutions of the equation
where , and when is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of , and . The behavior of solutions depends sensitively on whether is less, equal or bigger than the critical Sobolev exponent . For the solution asymptotically coincides with the solution of the equation in which the last term is absent. For the solution asymptotically coincides with the solution of the equation with . In the most delicate case the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on in a nontrivial way.
Cite this article
Vitaly Moroz, Cyrill B. Muratov, Asymptotic properties of ground states of scalar field equations with a vanishing parameter. J. Eur. Math. Soc. 16 (2014), no. 5, pp. 1081–1109DOI 10.4171/JEMS/455