Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation-Delta u + epsilon u - vertical bar u vertical bar(p-2) u + vertical bar u vertical bar(q-2) u = 0; x is an element of R-Nwhere N >= 3, q > p > 2 and epsilon > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p, q and N. The behaviour of solutions depends on whether p is less than, equal to or greater than the critical Sobolev exponent 2* = 2N/N-2. For p < 2* the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2* the solution asymptotically coincides with the solution of the equation with epsilon = 0. In the most delicate case p = 2* the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden-Fowler equation, whose choice depends on epsilon in a nontrivial way.