Abstract
This paper deals with semilinear elliptic problems of Dirichlet type, in star shaped domains. An abstract result is stated, which gives sufficient conditions for a positive solution of the problem to have strictly star shaped superlevels. Moreover, it turns out that the maximum point is the only critical point for these solutions. Then we apply the result to the "single-peak" solutions of some widely studied problems. First a nonlinear and subcritical elliptic equation is considered, when the nonlinearity approaches the critical one. Then Schrödinger type problems are studied. Finally, the case when the potential is constant is also analyzed, on a bounded domain.
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