JournalsaihpcVol. 24, No. 1pp. 41–60

Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

  • Giovanna Cerami

    Dipartimento di Matematica, Politecnico di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy
  • Riccardo Molle

    Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n○ 1, 00133 Roma, Italy
  • Donato Passaseo

    Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy
Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary cover
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Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the equation Δu+u=up1−\mathrm{\Delta }u + u = u^{p−1}, 2<p<2=2NN22 < p < 2^{ \ast } = \frac{2N}{N−2}, with Dirichlet zero data, in an unbounded smooth domain ΩRN\Omega \subset \mathbb{R}^{N} having unbounded boundary. Under the assumptions:

  • τ1,τ2,,τkR+{0}\exists \tau _{1},\tau _{2},…,\tau _{k} \in \mathbb{R}^{ + } \setminus \{0\}, 1kN21⩽k⩽N−2, such that

    (x1,x2,,xN)Ω    (x1,,xi1,xi+τi,,xN)Ω,i=1,2,,k,(x_{1},x_{2},…,x_{N}) \in \Omega \:\iff \:(x_{1},…,x_{i−1},x_{i} + \tau _{i},…,x_{N}) \in \Omega ,\:\forall i = 1,2,…,k,
  • RR+{0}\exists R \in \mathbb{R}^{ + } \setminus \{0\} such that RNΩ{(x1,x2,,xN)RN:j=k+1Nxj2R2}\mathbb{R}^{N} \setminus \Omega \subset \{(x_{1},x_{2},…,x_{N}) \in \mathbb{R}^{N}\text{:}\:\sum _{j = k + 1}^{N}x_{j}^{2}⩽R^{2}\}

the existence of at least k+1k + 1 solutions is proved.

Résumé

Dans cet article on étudie l'existence et la multiplicité de solutions positives pour l'équation Δu+u=up1−\mathrm{\Delta }u + u = u^{p−1}, 2<p<2=2NN22 < p < 2^{ \ast } = \frac{2N}{N−2}, avec la condition de Dirichlet u=0u = 0 sur ∂Ω. Le domaine ΩRN\Omega \subset \mathbb{R}^{N} est non borné et ∂Ω est non borné aussi. En supposant que les conditions

  • τ1,τ2,,τkR+{0}\exists \tau _{1},\tau _{2},…,\tau _{k} \in \mathbb{R}^{ + } \setminus \{0\}, 1kN21⩽k⩽N−2, tels que

    (x1,x2,,xN)Ω    (x1,,xi1,xi+τi,,xN)Ω,i=1,2,,k,(x_{1},x_{2},…,x_{N}) \in \Omega \:\iff \:(x_{1},…,x_{i−1},x_{i} + \tau _{i},…,x_{N}) \in \Omega ,\:\forall i = 1,2,…,k,
  • RR+{0}\exists R \in \mathbb{R}^{ + } \setminus \{0\} tel que RNΩ{(x1,x2,,xN)RN:j=k+1Nxj2R2}\mathbb{R}^{N} \setminus \Omega \subset \{(x_{1},x_{2},…,x_{N}) \in \mathbb{R}^{N}\text{:}\:\sum _{j = k + 1}^{N}x_{j}^{2}⩽R^{2}\}

soient vérifiées, on démontre que le problème possède au moins k+1k + 1 solutions.

Cite this article

Giovanna Cerami, Riccardo Molle, Donato Passaseo, Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 1, pp. 41–60

DOI 10.1016/J.ANIHPC.2005.09.007