We show the global structure of positive solutions for the Neumann problem involving mean curvature operator

where \(\lambda >0\) is a parameter, \(a:[0,R]\rightarrow {\mathbb {R}}\) is an \(L^1\)-function which is allowed to change sign and \(f:[0,\infty )\rightarrow [0,\infty )\) is continuous. Depending on the behavior of f near 0 and \(\infty \), we obtain that there exists \(0<\lambda _*\le \lambda ^*\) such that for any \(\lambda >\lambda ^*\), problem (P) possesses at least two positive solutions, while it has no solution for \(\lambda \in (0,\lambda _*)\). The proof of the main results is based upon bifurcation method.

我们展示了涉及平均曲率算子的 Neumann 问题正解的全局结构

其中\(\lambda >0\)是一个参数，\(a:[0,R]\rightarrow {\mathbb {R}}\)是一个\(L^1\) -允许改变符号的函数并且\(f:[0,\infty )\rightarrow [0,\infty )\)是连续的。根据f接近 0 和\(\infty \)的行为，我们得到存在\(0<\lambda _*\le \lambda ^*\)这样对于任何\(\lambda >\lambda ^* \)，问题 ( P ) 至少有两个正解，而\(\lambda \in (0,\lambda _*)\)没有解。主要结果的证明基于分岔法。