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Stable solutions for the bilaplacian with exponential nonlinearity. (English) Zbl 1138.35022

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.

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)
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