The Finite Element Approximation of Semilinear Elliptic Partial Differential Equations with Critical Exponents in the Cube

Abstract

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 $\lambda \lambda_{0}$ 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.