Uniqueness of positive solutions for a boundary blow-up problem
In this paper the author deals with the positive solutions to the boundary blow-up problem \[ \begin{cases} \Delta u=f(u) &\text{in }\Omega,\\ u=\infty &\text{on }\partial \Omega, \end{cases} \tag{1} \] where \(\Omega \) is a smooth bounded domain of \(\mathbb{R}^N\), \(f\) is a continuous function and a solution means a function \(u\in H_{\text{loc}}^1(\Omega )\cap C(\Omega )\) verifying the equation in \( (1)\) in the weak sense, and the boundary condition in the sense \(u(x)\rightarrow \infty \) as \(\text{dist}(x,\partial \Omega )\rightarrow 0\). First, the author shows that if \(f\) is increasing and there exists \(p>1\) such that \(\frac{f(t)}{t^p}\) is increasing for large \(t\), then \( (1)\) admits a unique positive solution. Then, they extend this uniqueness result to slightly more general problem than \((1)\), involving weights, namely \[ \begin{cases} \Delta u=a(x) f(u) &\text{in }\Omega,\\ u=\infty &\text{on }\partial\Omega, \end{cases} \] where \(a\in C(\overline{\Omega })\) is a nonnegative function, possibly vanishing on \(\partial \Omega\).
scientific article; zbMATH DE number 5626929
scientific article