Gelfand–Bratu problem

Remarks

This is a semilinear elliptic problem with a Dirichlet boundary condition; the exponential nonlinearity makes it a standard test case for bifurcation theory and numerical continuation. By the maximum principle every solution is positive in $\Omega$. There is a number $\lambda^* \in (0, \infty)$, depending on $\Omega$, such that a minimal solution exists for $0 < \lambda < \lambda^*$ and no solution exists for $\lambda > \lambda^*$. For $n = 1$ and $\Omega = (0, 1)$ the solutions are $u(x) = -2 \ln\big(\cosh(\vartheta (x - \tfrac12)/2) / \cosh(\vartheta/4)\big)$, where $\vartheta > 0$ solves $\vartheta = \sqrt{2 \lambda} \cosh(\vartheta/4)$. This equation has two roots for $\lambda < \lambda^*$, one for $\lambda = \lambda^*$ and none for $\lambda > \lambda^*$, with $\lambda^* \approx 3.5138$. On the unit ball of $\R^n$ with $n \ge 3$, the function $u = -2 \ln \abs{x}$ is a singular solution with $\lambda = 2(n-2)$. Joseph and Lundgren (1973) showed that on the unit ball with $3 \le n \le 9$ there are infinitely many solutions for $\lambda = 2(n - 2)$. In thermal explosion theory, the problem gives the steady states of the Frank-Kamenetskii model. In 1853 Liouville found the general solution of $\Delta u + \lambda e^{u} = 0$ in the plane, without boundary condition; this is a form of Liouville's equation.