A Class of Equations with Three Solutions

Here is one of the results obtained in this paper: Let Ω ⊂ R n be a smooth bounded domain, let q > 1 , with q < n + 2 n − 2 if n ≥ 3 and let λ 1 be the first eigenvalue of the problem − Δ u = λ u in Ω , u = 0 on ∂ Ω . Then, for every λ > λ 1 and for every convex set S ⊆ L ∞ ( Ω ) dense in L 2 ( Ω ) , there exists α ∈ S such that the problem − Δ u = λ ( u + − ( u + ) q ) + α ( x ) in Ω , u = 0 on ∂ Ω , has at least three weak solutions, two of which are global minima in H 0 1 ( Ω ) of the functional u → 1 2 ∫ Ω | ∇ u ( x ) | 2 d x − λ ∫ Ω 1 2 | u + ( x ) | 2 − 1 q + 1 | u + ( x ) | q + 1 d x − ∫ Ω α ( x ) u ( x ) d x where u + = max { u , 0 } .