Brézis, Haïm; Li, YanYan; Shafrir, Itai A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities. (English) Zbl 0794.35048 J. Funct. Anal. 115, No. 2, 344-358 (1993). Summary: Let \(u\) be a solution of the equation \(-\Delta u=V(x)e^ u\) in a domain \(\Omega\subset\mathbb{R}^ 2\), where \(0\leq a<V\leq b\) and \(V\) is Lipschitz continuous. We prove that sup \(u\) can be controlled in terms of \(\inf u\). More precisely, \(\sup_ Ku+\inf_ \Omega u\leq C(a,b,K,\Omega,\|\nabla V\|_{L^ \infty})\) for any compact subset \(K\subset\Omega\). This extends an earlier result of Shafrir who obtained a similar conclusion when \(V\equiv 1\). Cited in 5 ReviewsCited in 44 Documents MSC: 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B45 A priori estimates in context of PDEs Keywords:semilinear elliptic equation; exponential nonlinearity; supinf inequality PDFBibTeX XMLCite \textit{H. Brézis} et al., J. Funct. Anal. 115, No. 2, 344--358 (1993; Zbl 0794.35048) Full Text: DOI