The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p^su=λa(x)|u|+^(p-2)u+λb(x)|u|^(α-2)|u|~βu+μ(x)/αδ|u|^(γ-2)|v|~δu in Ω,(-△)_p^su=λc(x)|v|...The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p^su=λa(x)|u|+^(p-2)u+λb(x)|u|^(α-2)|u|~βu+μ(x)/αδ|u|^(γ-2)|v|~δu in Ω,(-△)_p^su=λc(x)|v|+^(q-2)v+λb(x)|u|~α|u|^(β-2)v+μ(x)/βγ|u|~γ|v|^(δ-2)v in Ω,u=v=0 on R^N\Ω where λ > 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 < p ≤ q < ∞, sq < N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ < λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.展开更多
基金supported by the National Natural Science Foundation of China(No.11771107)the Italian MIUR Project Variational Methods,with Applications to Problems in Mathematical Physics and Geometry(No.2015KB9WPT 009)+1 种基金the Gruppo Nazionale per l’Analisi Matematica,la Probabilitaè le loro Applicazioni(GNAMPA)of the Istituto Nazionale di Alta Matematica(INdAM)the INdAM-GNAMPA Project 2017 titled Equazioni Differenziali non lineari(No.Prot_2017_0000265)
文摘The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p^su=λa(x)|u|+^(p-2)u+λb(x)|u|^(α-2)|u|~βu+μ(x)/αδ|u|^(γ-2)|v|~δu in Ω,(-△)_p^su=λc(x)|v|+^(q-2)v+λb(x)|u|~α|u|^(β-2)v+μ(x)/βγ|u|~γ|v|^(δ-2)v in Ω,u=v=0 on R^N\Ω where λ > 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 < p ≤ q < ∞, sq < N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ < λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.
