Infinitely many positive solutions for a logarithmic fractional ℓ (⋅,⋅) -Kirchhoff problem

This paper is devoted to the study of the existence of infinitely many positive solutions for a class of fractional ℓ(⋅,⋅)-Kirchhoff-type problems involving a logarithmic nonlinearity. By employing a direct variational method and the framework of variable exponent fractional Sobolev spaces, we investigate the following nonlocal elliptic problem: {M(∫RN∫RN|ζ(σ)−ζ(τ)|ℓ(σ,τ)|σ−τ|N+sℓ(σ,τ)ln⁡(e+|ζ(σ)−ζ(τ)||σ−τ|s)dσ dτ∫RN∫RN)Lℓ(⋅,⋅),logsζ(σ)=θ(σ,ζ)in D,ζ=0on RN∖D, where D⊂RN is a smooth bounded domain and M denotes a Kirchhoff-type function. The operator Lℓ(⋅,⋅),logs represents the fractional ℓ(⋅,⋅)-Laplacian with logarithmic perturbation, defined for σ∈RN by Lℓ(⋅,⋅),logsζ(σ)=P.V.⁡∫RN|ζ(σ)−ζ(τ)|ℓ(σ,τ)−2 (ζ(σ)−ζ(τ))|σ−τ|N+sℓ(σ,τ)×[ℓ(σ,τ) ln⁡(e+|ζ(σ)−ζ(τ)||σ−τ|s)+|ζ(σ)−ζ(τ)|e|σ−τ|s+|ζ(σ)−ζ(τ)|]dτ. Here, the variable exponent ℓ:D×D→(1,∞) is continuous, s∈(0,1), and P.V. denotes the Cauchy principal value. Under appropriate structural assumptions on θ, we establish the existence of infinitely many distinct positive solutions whose L∞(D)-norms converge to zero.