11 research outputs found
Non-existence and uniqueness results for supercritical semilinear elliptic equations
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.Comment: Annales Henri Poincar\'e (2009) to appea
Stationary focusing mean-field games
We consider stationary viscous Mean-Field Games systems in the case of local,
decreasing and unbounded coupling. These systems arise in ergodic mean-field
game theory, and describe Nash equilibria of games with a large number of
agents aiming at aggregation. We show how the dimension of the state space, the
behavior of the coupling and the Hamiltonian at infinity affect the existence
and non-existence of regular solutions. Our approach relies on the study of
Sobolev regularity of the invariant measure and a blow-up procedure which is
calibrated on the scaling properties of the system. In very special cases we
observe uniqueness of solutions. Finally, we apply our methods to obtain new
existence results for MFG systems with competition, namely when the coupling is
local and increasing