Comparison results between linear and quasilinear elliptic boundary value problems (Q2276652)

One part of the paper concerns the following situation: Let \(\Omega =\Omega_ 0\setminus {\bar \Omega}_ 1\subset R^ n\) with \(\Omega_ 0\) and \(\Omega_ 1\) convex, let \(0\leq v\leq 1\) be a solution to \(-\Delta v=f(v)\) in \(\Omega\), \(v=1\) on \(\partial \Omega_ 1\), \(v=0\) on \(\partial \Omega_ 0\), \(f\in C\), let u be a solution to \(-div(a(| x|^ 2,u,| \nabla u|^ 2)\nabla u)=b(x,u,\nabla u)\) in \(\Omega\), \(u=\phi\) on \(\partial \Omega\). Conditions are given in order that \(v\geq u\) on \(\partial \Omega\) implies \(v\geq u\) in \(\Omega\). Another comparison concerns a general domain, homogeneous boundary data and the function independent of \(\nabla u\); various conditions on \(\Omega\) are also discussed, and, further, an asymptotic decay on infinite convex cylinders is investigated. (Related references for \(n=2,3\), and convex domains are \textit{C. O. Horgan} [J. Math. Anal. Appl. 107, 285-290 (1985; Zbl 0585.35004)], \textit{J. K. Knowles} [ibid. 59, 29-32 (1977; Zbl 0356.35032)], \textit{L. E. Payne} and \textit{G. A. Philippin} [Nonlinear Anal., Theory Methods Appl. 9, 787-797 (1985; Zbl 0526.35007)].