A class of functionals possessing multiple global minima

2020/11/24 by Ricceri, Biagio
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2011.12347

Abstract

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let Ω⊂ \bf Rⁿ (n≥ 2) be a smooth bounded domain and let Φ:\bf R²→ \bf R be a C¹ function, with Φ(0,0)=0, such that sup_(u,v)∈ \bf R²|Φ_u(u,v)|+|Φ_v(u,v)|\over 1+|u|^p+|v|^p0, with p=2\over n-2 when ngt;2. Then, for every convex set S⊆ L^∞(Ω)× L^∞(Ω) dense in L²(Ω)× L²(Ω), there exists (α,β)∈ S such that the problem\cases -Δu=(α(x)cos(Φ(u,v))-β(x)sin(Φ(u,v)))Φ_u(u,v) & in Ω \cr & \cr -Δv= (α(x)cos(Φ(u,v))-β(x)sin(Φ(u,v)))Φ_v(u,v) & in Ω \cr & \cr u=v=0 & on ∂Ω\crhas at least three weak solutions, two of which are global minima in H¹₀(Ω)× H¹₀(Ω) of the functional(u,v)→ 1\over 2 ( ∫_Ω|∇ u(x)|²dx+∫_Ω|∇ v(x)|²dx )-∫_Ω(α(x)sin(Φ(u(x),v(x)))+β(x)cos(Φ(u(x),v(x))))dx .

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