Summary
We consider the nonlinear Dirichlet problem
where Ω⊂R N is a bounded domain, u+=max{u,0} and u−=max{−u, 0}. The set of points (α, β) ε R2 for which (1) has nontrivial solutions is important for the study of certain nonlinear problems. It is the union of the graphs of some decreasing functions α→β(α). It is shown how the shape of the domain Ω influences the qualitative behaviour, such as convexity or concavity, of the functions β(.).
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D'Aujourd'hui, M. On the resonance set and the domain shape for a semilinear equation. Annali di Matematica pura ed applicata 159, 89–100 (1991). https://doi.org/10.1007/BF01766295
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DOI: https://doi.org/10.1007/BF01766295