Resumen de On a class of fractional Laplacian problems with variable exponents and indefinite weights

Nguyen Thanh Chung, Hoang Quoc Toan

• Let Ω⊂RN,N?2, be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form ⎧⎩⎨⎪⎪⎪⎪(Δ)sp1(x,.)u(x)+(Δ)sp2(x,.)u(x)+|u|q(x)−2u=λV1(x)|u(x)|r1(x)−2u(x)−μV2(x)|u(x)|r2(x)−2u(x) in Ω,u(x)=0 in ∂Ω, where (Δ)spi(.,.)(0?s?1),i=1,2, are the fractional pi(.,.)-Laplacians, pi∈C(Ω¯¯¯¯×Ω¯¯¯¯),q,ri∈C(Ω¯¯¯¯),i=1,2 while λ,μ are two positive parameters, V1,V2 are weight functions in generalized Lebesgue spaces Lα1(.)(Ω) and Lα2(.)(Ω) respectively such that V1 may change sign in Ω and V2(x)≥0 for all x∈Ω. Using variational techniques and Ekeland’s variational principle, we establish some existence results for the problem in an appropriate space of functions.