Energy functionals of Kirchhoff-type problems having multiple global minima. (March 2015) - Ricceri, Biagio
Abstract: In this paper, using the theory developed in Ricceri (2012), we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let Ω ⊂ R n be a smooth bounded domain, with n ≥ 4, let a, b, ν ∈ R, with a ≥ 0 and b > 0, and let p ∈ ] 0, n + 2 n − 2 [ . Then, for each λ > 0 large enough and for each convex set C ⊆ L 2 ( Ω ) whose closure in L 2 ( Ω ) contains H 0 1 ( Ω ), there exists v ∗ ∈ C such that the problem − ( a + b ∫ Ω | ∇ u ( x ) | 2 d x ) Δ u = ν | u | p − 1 u + λ ( u − v ∗ ( 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 corresponding energy functional.,