Ambrosetti-Prodi type result to a Neumann problem via a topological approach
The author studies the Neumann problem \[ \begin{aligned} u''+a(x)g(u) &=\mu+p(x), \\ u'(0)\;\;=\;\;u'(T)&=0, \end{aligned} \] where \(p\) is a bounded, measurable real function, \(g\) is \(C^1\) in \(\mathbb R\) with \(-\infty<g'(-\infty)<0<g'(+\infty)<+\infty\), and \(a\in L^\infty(0,T)\) is a non-negative function with \(\int_0^Ta(x)\,dx>0\). The main result says that if the number \(\mu\) is sufficiently large, the problem has at least two solutions. One of the solutions, say \(\tilde u\), is obtained by truncation and shooting and by construction it is negative (assuming, without loss of generality, that \(g(0)=0\) and \(g'(s)<0\) if \(s<0\)). A second solution is also obtained by shooting between the initial conditions of \(\tilde u+\varepsilon\), with \(\varepsilon>0\) small, and initial conditions \((u(0),0)\) with \(u(0)\) positive and large. The article contains a history of this kind of problems and stresses that a significant feature is the fact that the result holds with weight functions \(a\) that may vanish in a set of positive measure.
scientific article; zbMATH DE number 6810423
scientific article