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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
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