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

Posteraro, Maria Rosaria

C. R. Acad. Sci., Paris, Sér. I 322, No. 5, 445-450 (1996).

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 \(\Omega\).

MSC:

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

Keywords:

blow up at the boundary; semilinear elliptic equation; estimate of the minimum