Numerical and analytical estimates of existence regions for semi-linear elliptic equations with critical Sobolev exponents in cuboid and cylindrical domains
- Budd, C.J
- Humphries, A.R
DOIAbstract
AbstractWe 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 c...
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