## 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 $$\Omega$$.

### 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