Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness

Abstract

The paper concerns positive solutions for the Dirichlet problem{−Lu=ΛF(x,u)inΩ,u=0on∂Ω, where Ω is a smooth bounded domain in Rn, n≥2, u=(u1,…,um):Ω‾→Rm, m≥1, Lu=(L1u1,…,Lmum), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in Ω, Λ=(λ1,…,λm)∈Rm, F=(f1,…,fm):Ω×Rm→Rm and ΛF(x,u)=(λ1f1(x,u),…,λmfm(x,u)). For a general class of maps F we prove that there exists a hypersurface Λ⁎ in R+m:=(0,∞)m such that tuples Λ∈R+m below Λ⁎ correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above Λ⁎, there is no nonnegative strong solution. The shape of the hypersurface Λ⁎ depends on growth on u of the nonlinearity F in a sense to be specified. When Λ∈Λ⁎ and the coefficients of each operator Li are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all Li are Laplace operators, we investigate the L∞ regularity of this solution for any m≥1 in dimensions 2≤n≤9 for balls and n=2 and n=3 for convex domains.