Brézis, Haïm; Merle, Frank Uniform estimates and blow-up behavior for solutions of \(-\Delta{} u =V(x) e^ u\) in two dimensions. (English) Zbl 0746.35006 Commun. Partial Differ. Equations 16, No. 8-9, 1223-1253 (1991). The problem under investigation is (*) \(-\Delta u=fe^ u\) in \(\Omega\),\(u\mid_{\partial\Omega}=0\) in a bounded domain \(\Omega\)in \(R^ 2\), where \(f\in L^ p(\Omega)\) for some \(p\) in \(1<p\leq\infty\). If \(u\) is a solution of (*) with \(e^ u\in L^{p'}(\Omega)\), where \(p'\) denotes the conjugate exponent of \(p\), one result states that \(u\in L^ \infty(\Omega)\). A detailed investigation is given of the delicate question of uniform estimates for a sequence \(\{u_ n\}\) satisfying \(- \Delta u_ n=f_ n\exp u_ n\) in \(\Omega\),\(u_ n\mid_{\partial\Omega}=0\).Theorem. If \(f_ n\geq 0\) in \(\Omega\) and \(\| f_ n\|_ p\), \(\|\exp u_ n\|_{p'}\) are uniformly bounded, where \(\|\cdot\|_ p\) denotes the norm in \(L^ p(\Omega)\), then \(\{u_ n\}\) is bounded in \(L^ \infty_{loc}(\Omega)\). Also conditions are found for which \(\{u_ n\}\) is bounded in \(L^ \infty(\Omega)\), and examples are constructed with \(\| u_ n\|_ \infty\to\infty\) as \(n\to\infty\), i.e., the uniform estimate does not hold up to the boundary of \(\Omega\) in general. Reviewer: Charles A.Swanson (Vancouver) Cited in 14 ReviewsCited in 409 Documents MathOverflow Questions: A detail in one step in a theorem from a paper of Brezis and Merle Two doubts in the paper of Brezis Merle in blow up analysis of the equation \(-\Delta u=Ve^u\) MSC: 35B45 A priori estimates in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:exponential nonlinearity PDFBibTeX XMLCite \textit{H. Brézis} and \textit{F. Merle}, Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006) Full Text: DOI References: [1] Brezis H., Analyse Fonctionnelle [2] DOI: 10.1007/BF02392560 · Zbl 0636.53053 · doi:10.1007/BF02392560 [3] Chen W., Classification of solutions of some nonlinear elliptic equations 159 (1990) [4] John F., Comm. Pure Appl. Math. 14 pp 93– (1960) [5] DOI: 10.1007/BF02760233 · Zbl 0246.35025 · doi:10.1007/BF02760233 [6] K.Nagasaki ,T.Suzuki ,Asymptotic analysis for two dimensional elliptic eigenvalue problems with exponentially–dominated nonlinearities Asymptotic Analysis (to appear) · Zbl 0726.35011 [7] Suzuki T., Introduction to geometric potential theory 13 (1990) · Zbl 0754.31002 [8] S.Wang, An example of a blow–up sequence for This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.