This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫...This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.展开更多
基金supported by National Natural Science Foundation of China(11601515)Fundamental Research Funds for the Central Universities(3122017080)+3 种基金the second author acknowledges the support of the Slovenian Research Agency grants P1-0292,J1-8131,N1-0064,N1-0083,N1-0114the third author was supported by National Natural Science Foundation of China(11871199and 12171152)Shandong Provincial Natural Science Foundation,PR China(ZR2020MA006)Cultivation Project of Young and Innovative Talents in Universities of Shandong Province。
文摘This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator.More precisely,we study the following nonlocal problem:{M (∫∫_(R2N)|x|^(α1p)|y|^(α2p)|u(x) − u(y)|^(p)/|x − y|^(N+ps) dxdy)L_(p)^(s)u = |x| ^(β)f(u) in Ω,u = 0 in R^(N) \ Ω,where L_(p)^(s) is the generalized fractional p-Laplacian operator,N≥1,s∈(0,1),α_(1),α_(2),β∈R,Ω■R^(N) is a bounded domain with Lipschitz boundary,and M:R0^(+)→R0^(+),f:Ω→R are continuous functions.Firstly,we introduce a variational framework for the above problem.Then,the existence of least energy solutions is obtained by using variational methods,provided that the nonlinear term f has(θ_(p-1))-sublinear growth at infinity.Moreover,the existence of infinitely many solutions is obtained by using Krasnoselskii’s genus theory.Finally,we obtain the existence and multiplicity of solutions if f has(θ_(p-1))-superlinear growth at infinity.The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
