Existence and multiplicity results for <i>p</i> (⋅)& <i>q</i> (⋅) fractional Choquard problems with variable order
Jiabin Zuo, Alessio Fiscella, Anouar Bahrouni
Abstract
This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem (−Δ)p(⋅)s(⋅)u(x)+(−Δ)q(⋅)s(⋅)u(x)=λ|u(x)|β(x)−2u(x)+∫ΩF(y,u(y))|x−y|μ(x,y)dyf(x,u(x))+k(x)in Ω,u(x)=0in RN∖Ω, where (−Δ)p(⋅)s(⋅) and (−Δ)q(⋅)s(⋅) are two fractional Laplace operators with variable order s(⋅):R2N→(0,1) and with different variable exponents p(⋅):R2N→(1,∞) and q(⋅):R2N→(1,∞). Here Ω⊂RN is a bounded smooth domain with at least N≥2, λ is a real parameter, β, μ and k are continuous variable parameters, while F is the primitive function of a suitable f. Under some appropriate conditions on β and k, through variational methods, we prove existence and multiplicity of solutions for the above problem.