The formulation and the semismooth Newton solution of Nash equilibrium multiobjective elliptic optimal control problems are presented. Existence and uniqueness of a Nash equilibrium are proved. The corresponding solution is characterized by an optimality system that is approximated by second-order finite differences and solved with a semismooth Newton scheme. It is demonstrated that the numerical solution is second-order accurate and that the semismooth Newton iteration is globally and locally quadratically convergent. Results of numerical experiments confirm the theoretical estimates and show the effectiveness of the proposed computational framework.