Dávila, Juan; Dupaigne, Louis; Guerra, Ignacio; Montenegro, Marcelo Stable solutions for the bilaplacian with exponential nonlinearity. (English) Zbl 1138.35022 SIAM J. Math. Anal. 39, No. 2, 565-592 (2007). Summary: Let \(\lambda^*>0\) denote the largest possible value of \(\lambda\) such that \(\Delta^2 u= \lambda e^u\) in \(B\), \(u= \frac{\partial u}{\partial n}=0\) on \(B\}\) has a solution, where \(B\) is the unit ball in \(\mathbb R^N\) and \(n\) is the exterior unit normal vector. We show that for \(\lambda=\lambda^*\) this problem possesses a unique weak solution \(u^*\). We prove that \(u^*\) is smooth if \(N\leq 12\) and singular when \(N\geq 13\), in latter case \( u^*(r) = - 4 \log r + \log ( 8(N-2)(N-4) / \lambda^*) + o(1)\) as \(r\to 0\). We also consider the problem with general constant Dirichlet boundary conditions. Cited in 1 ReviewCited in 37 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) Keywords:bi-Laplacian; singular solutions; stability; exponential nonlinearity; Gelfand problem; Dirichlet condition PDFBibTeX XMLCite \textit{J. Dávila} et al., SIAM J. Math. Anal. 39, No. 2, 565--592 (2007; Zbl 1138.35022) Full Text: DOI arXiv Link