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

Abstract

We study the following Lane-Emden systemwith Ω a bounded regular domain of R N , N ≥ 4, and exponents p, q belonging to the so-called critical hyperbola 1/(p + 1) + 1/(q + 1) = (N − 2)/N . We show that, under suitable conditions on p, q, leastenergy (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. 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. If N ≥ 5, p = 1, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, if Ω is an annulus, we prove that least energy solutions are not radially symmetric, although there exist radial ones for any p, q > 0.