In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:
{(−Δ)α2a1u1(x)=uq111(x)+uq122(x)+h1(x,u1(x),u2(x),∇u1(x),∇u2(x)), x∈Ω,(−Δ)β2a2u2(x)=uq211(x)+uq222(x)+h2(x,u1(x),u2(x),∇u1(x),∇u2(x)), x∈Ω,u1(x)=0, u2(x)=0, x∈Rn∖Ω.
Here
(−Δ)α2a1 and
(−Δ)β2a2 denote weighted fractional Laplacians and
Ω⊂Rn is a
C2 bounded domain. It is shown that under some assumptions on
hi(i=1,2), the problem admits at least one positive solution
(u1(x),u2(x)). We first obtain the {a priori} bounds of solutions to the system by using the direct blow-up method of Chen, Li and Li. Then the proof of existence is based on a topological degree theory.