Energy functionals of Kirchhoff-type problems having multiple global minima

Original · EN

In this paper, using the theory developed in [8], we obtain some results of a totally new type about a class of non-local problems. Here is a sample: Let Ω⊂ Rⁿ 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 L²(Ω) whose closure in L²(Ω) contains H¹₀(Ω), there exists v*∈ C such that the problem - (a+b∫Ω|∇ u(x)|²dx)Δu =ν|u|ᵖ⁻¹u+λ(u-v*(x)) & in Ω & u=0 & on ∂Ω has at least three weak solutions, two of which are global minima in H¹₀(Ω) of the corresponding energy functional.