On multiple positive solutions of positone and non-positone problems (Q5930054)

This paper deals with the problem NEWLINE\[NEWLINE -\bigtriangleup u=f(u) \text{ in } \Omega, u=0 \text{ in } \Omega, NEWLINE\]NEWLINE where \(f\) is a locally Lipschitzian continuous function and \(\Omega\) is a ball in \(\mathbb{R}^N.\) NEWLINENEWLINENEWLINETwo classes of problems are considered: the positone problem \((f(0)\geq 0)\) and the non-positone problem \((f(0)<0).\) In the positone case, under some assumptions, it is proved that the above problem has at least three radial positive solutions with some properties. In the non-positone case, with several assumptions on \(f,\) it is proved that the considered problem has at least two radial positive solutions.