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A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities. (English) Zbl 0794.35048

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 \(V\equiv 1\).

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
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