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).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.