2022/11/09 by Pistoia, Angela, Schiera, Delia, Tavares, Hugo
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.04839
We study the following Lane-Emden system -Δu=|v|q-1v in Ω, -Δv=|u|p-1u in Ω, uν=vν=0 on ∂ Ω, with Ω a bounded regular domain of ℝN, 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, least-energy (sign-changing) solutions exist, and they are classical. 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, we prove some partial symmetry and symmetry-breaking results in the case Ω is a ball or an annulus.