Failure of an a priori estimate for a Dirichlet problem
Let \(\Omega\) be a bounded open set in \(\mathbb R^N\) (\(N\geq 3\)) with smooth boundary. The author is concerned with the study of positive solutions of the nonlinear elliptic equation \(-\Delta u=u^\alpha\) in \(\Omega\), under the Dirichlet condition \(u=\varepsilon\) on \(\partial\Omega\), where \(1<\alpha <(N+2)/(N-2)\) and \(\varepsilon >0\). The main result of the paper establishes that there exists \(\varepsilon^*>0\) depending on \(\Omega\) and \(\alpha\) such that the above problem admits a solution \(u_\varepsilon\in C^2(\Omega)\cap C(\overline\Omega )\) for all \(0\leq\varepsilon\leq\varepsilon^*\) and no solution exists if \(\varepsilon >\varepsilon^*\). In the particular case where \(\Omega\) is the unit ball in \(\mathbb R^N\) (\(N>3\)) it is obtained the lower bound \(\varepsilon^*\geq \left[N(N-2)/4\right]^{(N-2)/4}\). The proofs combine elliptic estimates with the maximum principle for elliptic equations.
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