The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents (Q2799616)

In this interesting paper, the author analyzes the problem NEWLINE\[NEWLINE\Delta^2 u = \frac{\lambda}{(1-u)^p}.NEWLINE\]NEWLINE He proves that any extremal solution is smooth (if \(p>1\) and \(N\leq 4\)) improving a result by \textit{Z. Guo} and \textit{J. Wei} [Discrete Contin. Dyn. Syst. 34, No. 6, 2561--2580 (2014; Zbl 1286.35090)], and any weak solution is smooth (if \(p=3\) and \(N=3\)) completing a result by \textit{J. Dávila} et al. [Math. Ann. 348, No. 1, 143--193 (2010; Zbl 1220.35047)]. Lastly, he studies the stability of the entire nonnegative solutions of \(\Delta^2 u =- \frac{1}{u^p}\) in \(\mathbb R^N\).