Numerical and analytical estimates of existence regions for semi-linear elliptic equations with critical Sobolev exponents in cuboid and cylindrical domains

摘要

We use a variety of careful numerical and semi-analytical methods to investigate two outstanding conjectures on the solutions of the parametrised semi-linear elliptic equation Δu+λu+u5=0,u>0, where u is defined to be zero on the boundary of a three dimensional domain. This equation is important in analysis and in studies of combustion and polytropic gases. It is known that there is a value λ0>0 such that no solutions exist for λ<λ0. McLeod and Schoen, and Bandle and Flucher have given different estimates for λ0; both of which have been conjectured to be exact. We perform a semi-analytic study of solutions on cylindrical domains and construct numerical approximations on cuboid domains using the finite element method combined with a careful post-processing step to reduce the otherwise significant errors. We compute these estimates for cylindrical and cuboid domains, and show that on these domains the estimates do not agree. We conclude that the conjecture of Bandle and Flucher is false. Our numerical computations on cuboid domains are consistent with McLeod's conjecture being true.