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

The paper concerns positive solutions for the Dirichlet problem & nbsp;{Lu = lambda F(x, u) in omega,u = 0 on & part;omega,& nbsp;where omega & nbsp;is a smooth bounded domain in R-n , n >=& nbsp;2, u = (u(1) , . . . , u(m)) : (omega)over bar ->& nbsp;R-m , m >=& nbsp;1, Lu = (L(1)u(1) , ... , L(m)u(m)), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in omega, lambda & nbsp;= (lambda(1) , . . . , lambda(m)) is an element of & nbsp;R-m , F = (f(1) , ... , f(m)) : omega & nbsp;x R-m & nbsp;->& nbsp;R-m and lambda F(x, u) = (lambda(1) f(1) (x, u) , ... , lambda(m)f(m)(x, u)). For a general class of maps F we prove that there exists a hypersurface lambda* in R-+(m)& nbsp;:= (0 , infinity)(m) such that tuples lambda & nbsp;is an element of & nbsp;R-+(m)& nbsp;below lambda* correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above lambda* , there is no nonnegative strong solution. The shape of the hypersurface lambda* depends on growth on u of the nonlinearity F in a sense to be specified. When & nbsp;lambda is an element of & nbsp;lambda* and the coefficients of each operator L-i & nbsp;are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all L-i are Laplace operators, we investigate the L-infinity & nbsp;regularity of this solution for any m >=& nbsp;1 in dimensions 2 <= n <=& nbsp;9 for balls and n = 2 and n = 3 for convex domains. (C)& nbsp;2022 Elsevier Inc. All rights reserved.