On the limiting problems for two eigenvalue systems

Abstract

Let Ω be a bounded, smooth domain. Supposing that α(p)+β(p)=p, ∀p∈(N,∞) and limp→∞⁡α(p)/p=θ∈(0,1), we consider two systems for the fractional p-Laplacian. The first one is given by{(−Δp)su(x)=λα(p)|u|α(p)−2u|v(x0)|β(p)inΩ,(−Δp)tv(x)=λβ(p)(∫Ω|u|α(p)dx)|v(x0)|β(p)−2v(x0)δx0inΩ,u=v=0inRN∖Ω, where x0 is a point in Ω‾, λ is a parameter, 0<s≤t<1, δx denotes the Dirac delta distribution centered at x and p>N/s. The second one is the system{(−Δp)su(x)=λα(p)|u(x1)|α(p)−2u(x1)|v(x2)|β(p)δx1inΩ,(−Δp)tv(x)=λβ(p)|u(x1)|α(p)|v(x2)|β(p)−2v(x2)δx2inΩ,u=v=0inRN∖Ω, where x1,x2∈Ω are arbitrary, x1≠x2. We obtain solutions for both systems and consider the asymptotic behavior of these solutions as p→∞.