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)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2...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)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2)^(q12)(x)+h_(1)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,(−Δ)_(a2)^(β/2)u2(x)=u_(1)^(q21)(x)+u_(2)^(q22)(x)+h_(2)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,u_(1)(x)=0,u_(2)(x)=0,x∈R^(n)∖Ω.Here(−Δ)_(a1)^(α/2) and(−Δ)_(a2)^(β/2) denote weighted fractional Laplacians andΩ⊂R^(n) is a C^(2) bounded domain.It is shown that under some assumptions on h_(i)(i=1,2),the problem admits at least one positive solution(u_(1)(x),u_(2)(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.展开更多
文摘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)_(a/1)^(α/2)u1(x)=u_(1)^(q11)(x)+u_(2)^(q12)(x)+h_(1)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,(−Δ)_(a2)^(β/2)u2(x)=u_(1)^(q21)(x)+u_(2)^(q22)(x)+h_(2)(x,u_(1)(x),u_(2)(x),∇u_(1)(x),∇u_(2)(x)),x∈Ω,u_(1)(x)=0,u_(2)(x)=0,x∈R^(n)∖Ω.Here(−Δ)_(a1)^(α/2) and(−Δ)_(a2)^(β/2) denote weighted fractional Laplacians andΩ⊂R^(n) is a C^(2) bounded domain.It is shown that under some assumptions on h_(i)(i=1,2),the problem admits at least one positive solution(u_(1)(x),u_(2)(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.
