On the solutions of the equation \(\Delta u=e^ u\) blowing up on the boundary. (English. Abridged French version) Zbl 0848.35037

Summary: Let \(\Omega\) be an open bounded set of \(\mathbb{R}^n\); we consider the equation \(\Delta u= e^u\) in \(\Omega\) with the boundary condition \(\lim_{x\to \partial\Omega} u(x)= +\infty\). We prove estimates for the solution \(u(x)\) and for the measure of \(\Omega\) comparing this problem with a problem of the same type defined in a ball. If \(n= 2\) we obtain an explicit estimate of the minimum of \(u(x)\) in terms of the measure of \(\Omega\).


35J60 Nonlinear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs