Parametric anisotropic double phase free boundary problems with nonlocal terms and convection: Existence, stability and asymptotic behavior

This paper is concerned with the study of a parametric anisotropic double phase free boundary problem (DPFBP, for short) involving $ (p(\cdot),q(\cdot)) $-Laplacian, convection term (a reaction depending on gradient), and two parameters (\theta,\lambda) appearing in q(\cdot) $-Laplacian term, respectively. The $-Laplace operator (DPFBP) considered to be controlled by highly nonlinear nonlocal functions. First, we apply surjectivity theorem pseudomonotone operators maximal monotone perturbation theory nonsmooth analysis examine nonemptiness compactness weak solution set (DPFBP). Then, explore asymptotic behavior (DPFBP), as \theta \lambda vary appropriate ranges. Finally, stability result provided, when obstacle function approximated convergent sequence.