On positive \(L^{\infty}\) solutions of a class of elliptic systems
Consider the weakly coupled elliptic system \(\ell_ 0\vec u=-\sum D_ i(a_{ij}D_ j\vec u)+\sum b_ jD_ j\vec u=\vec F(x,\vec u)\) in \(R^ m\), \(m\geq 3\). Under suitable growth conditions on \(\vec F\), we show the existence of infinitely many positive solutions which are bounded and bounded away from zero. Our main tools are a Harnack inequality some bounds on \(\ell_ 0\) in weighted \(L_ p\) spaces and Schauder's fixed point theorem. Our interest in this problem comes from a recent paper of \textit{N. Kawano} and \textit{T. Kusano} [Math. Z. 186, 287-297 (1984; Zbl 0523.34012)] where the case \(\ell_ 0=-\Delta\) and \(m=2\) was considered by means of suitable radial majorant conditions.
scientific article; zbMATH DE number 3925367