Existence of solutions for a class of fractional coupled Choquard–type systems with potential vanishing at infinity

This paper focuses on the following class of fractional coupled Choquard–type system{(−Δ)su1+V1(x)u1=(1|x|μ⁎F1(u1))f1(u1)+λ(x)u2,inRN,(−Δ)su2+V2(x)u2=(1|x|μ⁎F2(u2))f2(u2)+λ(x)u1,inRN,(u1,u2)∈Ds,2(RN)×Ds,2(RN),u1>0,u2>0,in RN, where (−Δ)s denotes the fractional Laplacian, s∈(0,1), N>2s, 0<μ<2s and Fi is the primitive of function fi. We consider continuous functions V1(x),V2(x) that may decay to zero at infinity and are related with the coupling function by λ(x)≤δmin⁡{V1(x),V2(x)}, for some δ∈(0,1/2). Based on variational methods, we prove existence of positive vector solutions by the penalization method.