Analysis of an elliptic system with infinitely many solutions

Abstract

We consider the elliptic system Δ⁢u=up⁢vq${\Delta u\hskip-0.284528pt=\hskip-0.284528ptu^{p}v^{q}}$, Δ⁢v=ur⁢vs${\Delta v\hskip-0.284528pt=\hskip-0.284528ptu^{r}v^{s}}$ in Ω with the boundary conditions ∂⁡u/∂⁡η=λ⁢u${{\partial u/\partial\eta}=\lambda u}$, ∂⁡v/∂⁡η=μ⁢v${{\partial v/\partial\eta}=\mu v}$ on ∂⁡Ω${\partial\Omega}$, where Ω is a smooth bounded domain of ℝN${\mathbb{R}^{N}}$, p,s>1${p,s>1}$, q,r>0${q,r>0}$, λ,μ>0${\lambda,\mu>0}$ and η stands for the outward unit normal. Assuming the “criticality” hypothesis (p-1)⁢(s-1)=q⁢r${(p-1)(s-1)=qr}$, we completely analyze the values of λ,μ${\lambda,\mu}$ for which there exist positive solutions and give a detailed description of the set of solutions.

Keywords

• elliptic system
• infinitely many solutions
• harnack inequality
• 35j47
• 35j60