On positive solutions of some singularly perturbed problems where the nonlinearity changes sign (Q1899889)

The existence of positive solutions of the Dirichlet problem \[ - \varepsilon \Delta u= g(u) \quad \text{in }D,\qquad u= 0 \quad \text{on } \partial D\tag{1} \] is studied. Here, \(D\) is a bounded domain in \(\mathbb{R}^n\) and \(g\) is \(C^1\)-function. Under various assumptions on \(D\) and \(g\) the author finds the exact number of positive solutions to (1). The results in this paper improve considerably analogous results of the author [Rocky Mt. J. Math. (to appear)].