The finite element approximation of semilinear elliptic partial differential equations with critical exponents in the cube

We consider the finite element solution of the parameterized semilinear ell iptic 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 import ant in analysis, and it is known that there is a value lambda(0) > 0 such t hat 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 t o be an estimate for its value. We then present results on computations on the full nonlinear problem. Using formal asymptotic methods we derive an ap proximate description of u which is supported by the numerical calculations . The asymptotic methods also give sharp estimates both for the error in th e finite element solution when lambda > lambda(0) and for the form of the s purious numerical solutions which are known to exist when lambda < lambda(0 ). These estimates are then used to post-process the numerical results to o btain a sharp estimate for lambda(0) which agrees with the conjectured valu e.