A Class of Equations with Three Solutions

Abstract

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 ⊆ H 0 1 ( Ω ) dense in H 0 1 ( 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 } .