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

Vitaly MorozCyrill B. Muratov — 2014

Journal of the European Mathematical Society

We study the leading order behaviour of positive solutions of the equation - Δ u + ϵ u - | u | p - 2 u + | u | q - 2 u = 0 , x N , where N 3 , q > p > 2 and when ϵ > 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 2 * = 2 N 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...

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