Blow-up analysis for solutions of \(-\Delta u = V e^ u\) in dimension two. (English) Zbl 0842.35011

The authors investigate the asymptotic behavior of a sequence of solutions to the second-order equation \(- \Delta u_n = V_n (x) e^{u_n}\) on a bounded domain \(\Omega \subset\mathbb{R}^2\). No boundary conditions are imposed upon the continuous solutions \(u_n\). The main theorem states, roughly, that if \(V_n\) is a nonnegative sequence that converges in \(C^0 (\overline \Omega)\) and \(e^{u_n}\) is bounded in \(L^1 (\Omega)\) for some sequence of solutions \(u_n\), then there exists a finite blow-up set \(S\).
The theorem is essentially an extension of a result of H. Brezis and F. Merle [Commun. Partial Differ. Equ. 16, No. 8/9, 1223-1253 (1991; Zbl 0746.35006)]. The main difference is a lack of a boundary condition. The main engine in the proof is the establishment of what the authors describe as a “sup+inf” inequality of the form \(\sup_K u + C_1 \inf_\Omega u \leq C_2\) for some compact set \(K \subset \Omega\).


35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations


Zbl 0746.35006
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