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On positive solutions of some singularly perturbed problems where the nonlinearity changes sign. (English) Zbl 0835.35013
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)].
Reviewer: P.Drábek (Plzeň)

##### MSC:
 35B25 Singular perturbations in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
exact number of positive solutions
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