Quantization for an elliptic equation of order 2m with critical exponential non-linearity |
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Authors: | Luca Martinazzi Michael Struwe |
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Institution: | 1. Centro di Ricerca Matematica E. De Giorgi, Scuola Normale Superiore, Pisa, Italy 2. ETH Zurich, Zurich, Switzerland
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Abstract: | On a smoothly bounded domain ${\Omega\subset\mathbb{R}^{2m}}$ we consider a sequence of positive solutions ${u_k\stackrel{w}{\rightharpoondown}0}$ in H m (Ω) to the equation ${(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}$ subject to Dirichlet boundary conditions, where 0 < λ k → 0. Assuming that $$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$ we prove that Λ is an integer multiple of Λ1 := (2m ? 1)! vol(S 2m ), the total Q-curvature of the standard 2m-dimensional sphere. |
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