EXISTENCE OF MULTIPLE SOLUTIONS FOR A FRACTIONAL p-LAPLACIAN SYSTEM WITH CONCAVE-CONVEX TERM

Abstract

Abstract: In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions

where Ω is a smooth bounded domain in Rⁿ, n > ps with s ∈ (0, 1) fixed, a(x), b(x), c(x) ≥ 0 and a(x), b(x), c(x) ∈ L^∞(Ω), 1< q < p and α, β > 1 satisfy p < α +β < p^*, p^*=np/n-ps. By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem (0.1).

Key words: fractional p-Laplacian system, Nehari manifold, multiple solutions

Junhui XIE, Xiaozhong HUANG, Yiping CHEN. EXISTENCE OF MULTIPLE SOLUTIONS FOR A FRACTIONAL p-LAPLACIAN SYSTEM WITH CONCAVE-CONVEX TERM[J].Acta mathematica scientia,Series B, 2018, 38(6): 1821-1832.

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References

[1] Ros-Oton X, Serra J. The Dirichlet problem for the fractional Laplacian:regularity up to the boundary. J Math Pures Appl, 2014, 10:275-302
[2] Ros-Oton X, Serra J. The Pohozaev identity for the fractional Laplacian. Arch Ration Mech Anal, 2014, 213:587-628
[3] Servadei R, Valdinoci E. Mountain pass solutions for non-local elliptic operators. J Math Anal Appl, 2012, 389:887-898
[4] Servadei R, Valdinoci E. The Brezis-Nirengerg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367:67-102
[5] Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136:521-573
[6] Bucur C, Valdinoci E. Non-Local Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana 20. Springer, 2016
[7] Chen W, Deng S. The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal, 2016, 27:80-92
[8] Wei Y, Su X. Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian. Calc Var Partial Differential Equations, 2015, 52:95-124
[9] Guo Z, Luo S, Zou W. On critical sysyems involving fractional Laplacian. J Math Anal Appl, 2017, 446:681-706
[10] Goyal S, Sreenadh K. A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function. Proc Indian Acad Sci, 2015, 125:545-558
[11] Drábek P, Pohozaev S I. Positive solutions for the p-Laplacian:Application of the fibering method. Rov Soc Edinburgh Sect A, 1997, 127:703-726
[12] Brown K J, Zhang Y. The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J Differ Equ, 2003, 193:481-499
[13] Brezis H. Analyse Fonctionelle. Théorie et Applications. Paris:Masson, 1983
[14] Chen W, Deng S. The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities. Z Angew Math Phys, 2015, 66:1387-1400
[15] W Chen, Xie J. Multiple solutions to the nonhomogeneous Kirchhoff type problem involving a nonlocal operator. Diff Eq Appl, 2016, 8:367-375
[16] Xu Y, Tan Z, Sun D. Multiplicity results for a nonlinear ellptic problem involving the fractional Laplacian. Acta Math Sci, 2016, 36B:1793-1803