Asymptotic properties of ground states of scalar field equations with a vanishing parameter
Abstract
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 $p<p^\ast$ the solution asymptotically coincides with the solution of the equation 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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2012
- DOI:
- 10.48550/arXiv.1202.3426
- arXiv:
- arXiv:1202.3426
- Bibcode:
- 2012arXiv1202.3426M
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- Nonlinear Sciences - Pattern Formation and Solitons
- E-Print:
- J. Eur. Math. Soc. 16, 1081-1109 (2014)