Entire solutions of quasilinear symmetric systems

الأصل · EN

We study the following quasilinear elliptic system for all i=1,,m equation* -div(Φ'(|∇ uᵢ|²) ∇ uᵢ) = Hᵢ(u) in Rⁿ equation* where u=(uᵢ)ᵢ₌₁ᵐ: Rⁿ→ Rᵐ and the nonlinearity Hᵢ(u) ∈ C¹(Rᵐ)→ R is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the p-Laplacian operators fit in the above form, for appropriate Φ. We establish a Hamiltonian identity of the following form for all xₙ equation* ∫ᵣⁿ⁻¹ (∑ᵢ₌₁ᵐ [1/2 Φ(|∇ uᵢ|²) - Φ'(|∇ uᵢ|²) |∂ₓₙ uᵢ|²] - H(u)) d x'≡ C, equation* where x=(x',xₙ)ⁿ and H is the antiderivative of H=(Hᵢ)ᵢ₌₁ᵐ. This can be seen as a counterpart of celebrated pointwise inequalities provided by Caffarelli, Garofalo and Segala in cgs and by Modica in m. For the case of system of equations, that is when m≥ 2, we show that as long as α≥ α*:=ₛ>₀{2 s Φ'(s)/Φ(s)} the function Iα(r):=1rⁿ⁻α ∫Bᵣ ∑ᵢ₌₁ᵐ Φ(|∇ uᵢ|²) - 2 H(u) is monotone nondecreasing in r. We call this a weak monotonicity formula since for m=1 it is shown in cgs that Iα(r) is monotone when α≥ 1, under certain conditions on Φ. We prove De Giorgi type results and Liouville theorems for H-monotone and stable solutions in two and three dimensions when the system is symmetric.