The finite element approximation of semi-linear elliptic partial difference equations with critical exponents in the cube

We consider the finite element solution of the parameterized semilinear elliptic equation $\Delta u + \lambda u + u^{5} = 0, u > 0$, where u is defined in the unit cube and is 0 on the boundary of the cube. This equation is important in analysis, and it is known that there is a value $\lambda_{0} > 0$ such that no solutions exist for $\lambda < \lambda_{0}$. By solving a related linear equation we obtain an upper bound for $\lambda_{0}$ which is also conjectured to be an estimate for its value. We then present results on computations on the full nonlinear problem. Using formal asymptotic methods we derive an approximate description of u which is supported by the numerical calculations. The asymptotic methods also give sharp estimates both for the error in the finite element solution when $\lambda > \lambda_{0}$ and for the form of the spurious numerical solutions which are known to exist when $\lambda < \lambda_{0}$. These estimates are then used to post-process the numerical results to obtain a sharp estimate for $\lambda_{0}$ which agrees with the conjectured value.