Archiv der Mathematik

, Volume 91, Issue 4, pp 354–365 | Cite as

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

Article

Abstract.

In this paper we consider the Lane–Emden problem adapted for the p-Laplacian
$$\left\{\begin{array}{l} -\Delta_{p}u = \lambda |u|^{q-2}u,\quad {\rm in}\,\Omega,\\ \quad\quad u=0, \quad\quad\quad\quad\,{\rm on}\, \partial \Omega,\end{array}\right.$$
where Ω is a bounded domain in \(\mathbb{R}^n\), n ≥ 2, λ > 0 and p < qp* (with \(p^* = \frac{np}{n-p}\) if p < n, and p* = ∞ otherwise). After some recalls about the existence of ground state and least energy nodal solutions, we prove that, when qp, accumulation points of ground state solutions or of least energy nodal solutions are, up to a “good” scaling, respectively first or second eigenfunctions of  −Δp.

Keywords.

p-Laplacian energy functional ground state solutions least energy nodal solutions (nodal) Nehari manifold first and second eigenfunctions of  −Δp 

Mathematics Subject Classification (2000).

Primary 35J20 Secondary 35P30, 35J70, 49J40 

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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Institut de MathématiqueUniversité de Mons-HainautMonsBelgium
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany

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