Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order. Issue 2 (1st February 2022) - Zuo, Jiabin; Fiscella, Alessio; Bahrouni, Anouar
Abstract : This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem 1 ( − Δ ) p ( ⋅ ) s ( ⋅ ) u ( x ) + ( − Δ ) q ( ⋅ ) s ( ⋅ ) u ( x ) = λ | u ( x ) | β ( x ) − 2 u ( x ) + ∫ Ω F ( y, u ( y ) ) | x − y | μ ( x, y ) d y f ( x, u ( x ) ) + k ( x ) i n   Ω, u ( x ) = 0 i n   R N ∖ Ω, where ( − Δ ) p ( ⋅ ) s ( ⋅ ) and ( − Δ ) q ( ⋅ ) s ( ⋅ ) are two fractional Laplace operators with variable order s ( ⋅ ) : R 2 N → ( 0, 1 ) and with different variable exponents p ( ⋅ ) : R 2 N → ( 1, ∞ ) and q ( ⋅ ) : R 2 N → ( 1, ∞ ) . Here Ω ⊂ R N 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.