The singular extremal solutions of the bi-Laplacian with exponential nonlinearity
Amir Moradifam · Proceedings of the American Mathematical Society · 2009
Consider the problem { Δ 2 u = λ e u a m p ; in B , u = ∂ u ∂ n = 0 a m p ; on ∂ B , \begin{eqnarray*} \left \{ \begin {array}{ll} \Delta ^2 u= \lambda e^{u} &\text {in } B,\\ u=\frac {\partial u}{\partial n}=0 &\text {on }\partial B, \end{array} \right . \end{eqnarray*} where B B is the unit ball in R N {\mathbb {R}}^N and λ \lambda is a parameter. Unlike the Gelfand problem the natural candidate u = − 4 ln ( | x | ) u=-4\ln (|x|) , for the extremal solution, does not satisfy the boundary conditions, and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Recently a computer-assisted proof was used to show that the extremal solution is singular in dimensions 13 ≤ N