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

In this article, we study the existence of multiple solutions for the following sys-tem driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions {（-△）p^su=a（x）｜u｜^q-2u＋2α/α＋βc（x）｜u｜^α-2u｜u｜^β,in Ω,（-△）p^su=a（x）｜u｜^q-2u＋2α/α＋βc（x）｜u｜^α｜u｜^β-2.inΩ,（0.1）u=v=0,in R^n／Ωwhere Ω is a smooth bounded domain in R^n, n 〉 ps with s ∈ （0, 1） fixed, a（x）, b（x）, c（x） 〉 0and 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 ofsolutions to problem （0.1）.

Abstract In this article, we study the existence of multiple solutions for the following sys-tem driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions {（-△）p^su=a（x）｜u｜^q-2u＋2α/α＋βc（x）｜u｜^α-2u｜u｜^β,in Ω,（-△）p^su=a（x）｜u｜^q-2u＋2α/α＋βc（x）｜u｜^α｜u｜^β-2.inΩ,（0.1）u=v=0,in R^n／Ωwhere Ω is a smooth bounded domain in R^n, n 〉 ps with s ∈ （0, 1） fixed, a（x）, b（x）, c（x） 〉 0and 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 ofsolutions to problem （0.1）.