On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

Abstract

We study positive solutions to the fractional Lane-Emden system ⎧⎩⎨⎪⎪(−Δ)su(−Δ)svu=v=vp+μ=uq+ν=0inΩinΩinΩc=ℝN∖Ω,(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, p > 0, q > 0 and μ, ν are positive measures in Ω. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν. Furthermore, if p, q ∈ (1,N+sN−s) and 0 ≤ μ, ν ∈ Lr(Ω), for some r > N2s, we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.