Nonradial Solutions of a Nonlinear Elliptic Equation

Abstract: We consider the partial differential equation ∆u + f(u) = 0 in two dimensions,where f is a nonlinear function of u. Under certain assumptions on f we show that the equation has non-radial C² solutions with u(x) → 0 as |x| → 0 with an arbitrary number of zeros on each axis. The equation has applications in Bose-Einstein condensation and water waves among other fields. We also look at numerical solutions in the special case f(u) = |u|²u − u.