Dirichlet problem for a nonlocal p-Laplacian elliptic equation

The paper deals with a nonlinear nonlocal second-order p-Laplacian equation with Dirichlet boundary conditions. A rigorous proof for existence and uniqueness of the weak solution is presented. The weak formulation of the problem of interest is transformed into an unconstrained minimization problem. A variational inequality for the objective functional is obtained. A priori estimates for the weak solution are proved. The optimization problem is solved by means of the two-point step size gradient method. A steplength for the iterative method assuring monotone decrease of the error in approximate solutions is found. Q-linear convergence of the finite element approximations to the discrete solution is established. A monotone error reduction is proved. An original procedure for obtaining initial guesses is developed. Numerical examples supporting the developed theory are discussed.