Fractional Logarithmic Double Phase Problems: Qualitative Analysis in the Anisotropic Case

Abstract. This paper is concerned with the study of elliptic differential problems involving fractional variable exponent double phase operators with logarithmic perturbation [Formula: see text] generated by [Formula: see text]. In the first part, we study fractional double phase elliptic inclusions with a generalized multivalued mapping and a maximal monotone operator which is formulated by the convex subdifferential of the indicator function to a convex set. Based on the sub-supersolution method along with truncation techniques and nonsmooth analysis we show an existence result and give an application construction such a pair of sub-supersolution. Additionally, under lattice conditions, we establish the compactness and the directedness of the solution set within a pair of sub- and supersolutions. In the second part, we consider a type of fractional Kirchhoff double phase problems governed by the operator [Formula: see text]. Applying variational methods, the Poincaré–Miranda existence theorem together with the quantitative deformation lemma, we prove a multiplicity result which says that the problem has at least a positive solution, a negative solution, and a sign-changing solution.