We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let ##IMG## [http://ej.iop.org/images/1064-5632/84/3/592/IZV_84_3_592ieqn1.gif] {$\widetilde{u}(x)$} be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting ##IMG## [http://ej.iop.org/images/1064-5632/84/3/592/IZV_84_3_592ieqn2.gif] {$v(x)=u(x)-\widetilde{u}(x)$} , we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function ##IMG##

中文翻译：

---

一类带参数和不连续非线性的椭圆型边值问题

我们研究了具有不均匀Dirichlet条件，不连续非线性和正参数作为非线性因素的有界域中的椭圆边值问题。非线性位于方程的右侧。对于相位变量的负（resp。，非负）值，它为非正（resp。等于零）。让## IMG ## [http://ej.iop.org/images/1064-5632/84/3/592/IZV_84_3_592ieqn1.gif] {$ \ widetilde {u}（x）$}是以下解决方案：右边为零的边界值问题（假定边界函数为正）。放入## IMG ## [http://ej.iop.org/images/1064-5632/84/3/592/IZV_84_3_592ieqn2.gif] {$ v（x）= u（x）-\ widetilde {u} （x）$}，我们将原始问题简化为具有齐次边界条件的问题。