Non-existence and uniqueness results for supercritical semilinear elliptic equations

Jean Dolbeault (1) , Robert Stanczy (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Instytut Matematyczny

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

Keywords: Semi-linear elliptic equation; uniqueness; Gelfand's problem; Pohozaev's method; Rellich-Pohozaev identities; Brezis-Nirenberg problem; bifurcation; branches of solutions; non-existence results; critical explosion parameter; nonlinear eigenvalue problem; non-local constraints

Domains: Analysis of PDEs [math.AP]

Jean Dolbeault, Robert Stanczy. Non-existence and uniqueness results for supercritical semilinear elliptic equations. Annales Henri Poincaré, 2009, 10 (7), pp.1311-1333. ⟨hal-00349574v2⟩