Determine the threshold parameter tilde-lambda for the cube-domain Bratu equation

Determine the exact values of the threshold parameter \tilde{\lambda} for each spatial dimension d for the Bratu equation with zero Dirichlet boundary conditions on the cube domain [0,1]^d, where the equation is \Delta u + \lambda e^{u} = 0, such that a countable infinity of solutions occurs at \lambda = \tilde{\lambda} (analogous to the Joseph–Lundgren threshold 2(d−2) for the ball domain).

Background

The paper presents numerical evidence that the bifurcation structure of the Bratu equation on cube domains [0,1]^d closely resembles the well-known Joseph–Lundgren description on ball domains. For 3 ≤ d ≤ 5, their simulations suggest the existence of a special parameter \tilde{\lambda} at which there are countably many solutions, mirroring the ball-domain result where \tilde{\lambda} = 2(d−2).

However, while the qualitative behavior appears similar, the authors do not identify the actual values of \tilde{\lambda} for cube domains. They explicitly state that these values remain unknown, highlighting a concrete unresolved question needed to complete the analogy with the ball-domain theory.