Multiplicity Results Near Resonance For Some Elliptic Type Equation And System Of Fractional And Non-local Operators

In this paper, some elliptic type equation and system of fractional and non-local operators are studied by a local saddle point theorem, the Linking Theorem in critical point theory and analysis techniques. We obtain at least two solutions in the case of near resonance at higher eigenvalues. The results improve and generalize some of the corresponding existing results. More precisely, we consider the following elliptic equation and system where Ω(?)RN(N≥1) is an open bounded domain with Lipschitz boundary(?)Ω.LKu may be denned as LKu =1/2fRn (u(x + y)+ u(x -y)- 2u(x))K(y)dy,x∈Rn h(x)∈L2(Ω),and the nonlinear term f(x,u) :Ω×Râ†'R is a Caratheodory function which satisfies:(f1):lim|t|â†'∞f(x,t)/t=0,uniformly with respect to x∈Ω,and (?)M >0, there exists gM(x) ∈L2(Ω),such that (?)x∈Ω,|t|≤M,|f(x,t)|≤gM(x). where Ω(?)RN(N≥1) is an open bounded domain with Lipschitz boundar(?)Ω.LKiu denned as follows: LKiu=1/2f (u(x + y)+u(x-y)-2u(x))Ki(y)dy,x∈Rn,i =l,2Here Ki:Rn\{0}â†'(0,+∞) is a function such that mKi∈L1(Rn ,m(x) min{|x|2,1},there exists θ > 0 and s∈ (0,1) such that Ki(x)≥θ|x|-(n+2x),for any x∈Rn\{0} and Ki(x) = Ki(-x) for any x∈Rn\{0}.F ∈ C1(Ω×R2,R) verifying the following conditions: lim|â–½F(x,s)|/|s|=0 uniformly x∈Ω where â–½F = (Fu,Fv) denotes the gradient of F with respect to(u,v) ∈ R2.