A Scattering Operator for Some Nonlinear Elliptic Equations

We consider nonlinear elliptic equations of the form Δ u = f(u,∇ u) for the suitable analytic nonlinearity f, in the vinicity of infinity in ℝ^d , which is on the complement of a compact set. We show that there is a one-to-one correspondence between the nonlinear solution u defined there and the linear solution u_L to the Laplace equation such that, in an adequate space, u - u_L→ 0 as |x|→ +∞ . This is a kind of scattering operator. Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the H-system. Similar results are derived for solutions defined on the neighborhood of a point in ℝ^d . The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).