We consider semilinear elliptic boundary value problems of the form \[Lu = λ f(x,u)\] with Dirichlet boundary conditions. By using variational methods, we show how changes in the sign of f(x, · ) lead to multiple positive solutions of the equation for sufficiently large λ. In addition more detailed results are obtained for autonomous ordinary differential equations by using simple quadrature arguments.
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Brown et al. (1979) studied this question.