Differential inclusion systems with fractional competing operator and multivalued fractional convection term
Jinxia Cen, Salvatore A. Marano, Shengda Zeng
TL;DR
The paper studies a fractional differential inclusion system on a bounded domain $\Omega$ with operators $(-\Delta)^{s_i}_{p_i}$ and $(-\Delta)^{t_i}_{q_i}$ and a fractional convection term $D^{r_i}u_i$, under homogeneous Dirichlet data. Due to potential lack of ellipticity or monotonicity when $\mu<0$, standard variational methods are inapplicable, motivating a Galerkin approach combined with a finite-dimensional surjectivity theorem for multifunctions to obtain generalized solutions. Existence is proved under a set of hypotheses $({\rm H}_1)$–$({\rm H}_4)$ with a smallness condition, with a stronger growth condition yielding a strongly generalized solution, and nonnegative $\mu_i$ granting a weak solution. The results extend the theory of nonlocal, multivalued fractional inclusions with competing operators and Dirichlet boundary conditions, providing a robust existence framework for fractional convection–reaction systems.
Abstract
In this work, the existence of solutions (in a suitable sense) to a family of inclusion systems involving fractional, possibly competing, elliptic operators, fractional convection, and homogeneous Dirichlet boundary conditions is established. The technical approach exploits Galerkin's method and a surjective results for multifunctions in finite dimensional spaces as well as approximating techniques.