Nondegeneracy and uniqueness for boundary blow-up elliptic problems
The aim of the author is to show uniqueness for \[ \Delta u=\lambda a(x) u^p+ g(x,u)\quad\text{in }\Omega, \] \[ u=+\infty\quad\text{on }\partial\Omega, \] where \(p> 1\), \(a(x)\) is a continuous function which can be singular near \(\partial\Omega\), \(g(x,u)\) is a prescribed continuous function and \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\). The author determines asymptotic behaviour this unique solution for large enough \(\lambda\). The author considers more general problem, obtained by replacing \(u^p\) by \(e^u\) or ``logistic type'' function \(f(u)\). The main tool of the author is the implicit function theorem.
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