3 years ago

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

Mousomi Bhakta, Phuoc-Tai Nguyen
We study positive solutions to the fractional Lane-Emden system (−Δ)su=vp+μin Ω(−Δ)sv=uq+νin Ωu=v=0in Ωc=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, νLr(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. Keywords: nonlocal; system; existence; multiplicity; linking theorem; measure data; source terms; positive solution MSC 2010: Primary 35R11; 35J57; 35J50; 35B09; 35R06
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