JournalsjemsVol. 16, No. 5pp. 1081–1109

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

  • Vitaly Moroz

    University of Wales Swansea, UK
  • Cyrill B. Muratov

    New Jersey Inst. of Technology, Newark, USA
Asymptotic properties of ground states of scalar field equations with a vanishing parameter cover
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Abstract

We study the leading order behaviour of positive solutions of the equation

Δu+ϵuup2u+uq2u=0,xRN,-\Delta u +\epsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\mathbb R^N,

where N3N\ge 3, q>p>2q>p>2 and when ϵ>0\epsilon >0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of pp, qq and NN. The behavior of solutions depends sensitively on whether pp is less, equal or bigger than the critical Sobolev exponent 2=2NN22^\ast=\frac{2N}{N-2}. For p<2p<2^\ast the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p>2p>2^\ast the solution asymptotically coincides with the solution of the equation with ϵ=0\epsilon =0. In the most delicate case p=2p=2^\ast the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on ϵ\epsilon 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. 1081–1109

DOI 10.4171/JEMS/455