The paper deals with the existence of positive solutions of the problem - Delta u = u(p) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain of R-n, n >= 3, and p > 2. We describe new concentration phenomena, which arise as p-->+infinity and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p --> +infinity, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Omega as p --> +infinity. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains ( while no solution can exist if Omega is starshaped and p >= 2n/n-2, as a consequence of the Pohozaev's identity).

Molle, R., Passaseo, D. (2006). Nonlinear elliptic equations with large supercritical exponents. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 26(2), 201-225 [10.1007/s00526-005-0364-3].

Nonlinear elliptic equations with large supercritical exponents

MOLLE, RICCARDO;
2006-01-01

Abstract

The paper deals with the existence of positive solutions of the problem - Delta u = u(p) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain of R-n, n >= 3, and p > 2. We describe new concentration phenomena, which arise as p-->+infinity and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p --> +infinity, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Omega as p --> +infinity. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains ( while no solution can exist if Omega is starshaped and p >= 2n/n-2, as a consequence of the Pohozaev's identity).
2006
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Concentration phenomena; Domain geometry; Supercritical exponents
Molle, R., Passaseo, D. (2006). Nonlinear elliptic equations with large supercritical exponents. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 26(2), 201-225 [10.1007/s00526-005-0364-3].
Molle, R; Passaseo, D
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/38910
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