Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions.

Abstract: I will present some recent results obtained in collaboration with A. Pistoia and H. Tavares for a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and critical nonlinearities. We show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem, and we establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. I will also briefly present existence of least-energy solutions for the particular case in which the system reduces to a biharmonic equation, and some symmetry results in the case the domain is an annulus.

Posted on Dec 5, 2022 in Differential Equations, Seminars