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