Some results concerning the Poisson-Boltzmann equation. (English) Zbl 0756.35029
The authors study the nonlinear elliptic boundary value problem
\[
\Delta u+\sigma\cdot\mu\cdot\exp u=0,\quad u\mid_{\partial\Omega}=0,
\]
where \(\mu=\left(\int_ \Omega\exp u\cdot dV\right)^{-1}\) and \(\sigma\) is a positive constant. This problem describes the gravitational potential of self-gravitating thermodynamically equilibrium gas filling up a bounded domain \(\Omega\subset\mathbb{R}^ 3\). Using the Schauder techniques, it is shown, that the problem has a solution for all sufficiently small \(\sigma\) and has no solution, if \(\sigma>3\left(\int_{\partial\Omega}{dS\over(x,n)}\right)^{-1}\), where \(n\) denotes the exterior unit normal to \(\partial\Omega\).
Reviewer: O.Titow (Berlin)
MSC:
35J60 | Nonlinear elliptic equations |
35J25 | Boundary value problems for second-order elliptic equations |
82D05 | Statistical mechanics of gases |
82B30 | Statistical thermodynamics |