Special Session 140: Symmetry and Overdetermined problems

The quasi-linear Liouville equation

Pierpaolo Esposito Universit\\' degli Studi Roma Tre Italy Co-Author(s):

Abstract: In R^n we discuss classification and/or quantization results for a quasi-linear Liouville equation involving the n-Laplace operator and an exponential nonlinearity with the possible presence of some singular sources.

Overdetermined problems for p-Laplace and generalized Monge-Ampere equations

Yichen Liu Xi'an Jiaotong-Liverpool University Peoples Rep of China Co-Author(s):    Behrouz Emamizadeh, Giovanni Porru

Abstract: We investigate overdetermined problems for p-Laplace and generalized Monge Ampere equations. By using the theory of domain derivative, we find duality results and characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain.

Break of symmetry for semilinear elliptic problems in cones

Filomena Pacella University of Roma Sapienza Italy Co-Author(s):    Giulio Ciraolo and Camilla Polvara

Abstract: We consider semilinear elliptic problems with mixed boundary conditions in spherical sectors contained in an unbounded cone in $\mathbb{R^N}$ and address the question of the radial symmetry of the positive solutions. We present some results which show that the symmetry, as well as the break of symmetry depends on the kind of cones considered. This implies that a Gidas-Ni-Nirenberg type result does not hold in any spherical sectors. Similar results hold for the critical exponent Neumann problem in the whole cone.

Elliptic systems with critical growth

Angela Pistoia Sapienza University of Roma Italy Co-Author(s):

Abstract: I will present some old and new results concerning existence of sign-changing solutions to the Yamabe problem on the round sphere and existence of positive solutions to a class of systems of PDE`s with critical growth in the whole euclidean space in presence of a competitive regime.

On a Bliss-Moser type inequality

Bernhard Ruf Accademia di Scienze e Lettere - Istituto Lombardo Italy Co-Author(s):

Abstract: We derive a limiting inequality for the integral inequalities by Bliss. We then consider a critical version of this inequality which is of Moser type, and discuss related non-compactness properties. Furthermore, we show that this inequality is related to critical boundary growth for functions on a disk in two dimensions. Finally, we prove the existence of solutions for related critical boundary value problems.