NONEXISTENCE OF SOLUTIONS AND AN ANTI-MAXIMUM PRINCIPLE FOR COOPERATIVE SYSTEMS WITH THE P-LAPLACIAN

We obtain a nonexistence result and an anti-maximum principle for weak solutions u = (u(1),...,u(n)) to the following strictly cooperative e lliptic system, [GRAPHICS] Here, Omega subset of IRN is a bounded doma in with a C-2,C-alpha-boundary partial derivative Omega, for some alph a is an element of (0,1), Delta(p) denotes the p-Laplacian defined by Delta(p)u (sic) div (\del u/(p-2)del u) for p is an element of (1,infi nity), and the coefficients a(ij) (1 less than or equal to i, j less t han or equal to n) are assumed to be constants satisfying a(ij) > 0 fo r i not equal j (a strictly cooperative system). We assume 0 less than or equal to f(i) is an element of L-infinity(Omega) (1 less than or e qual to i less than or equal to n). For Lambda(+) = Lambda(-) = Lambda is an element of IR and f = (f(1),...,f(n)) = 0 in Omega, let mu(1) d enote the first eigenvalue Lambda of the (p - 1) - homogeneous system (S). Assuming f not equivalent to 0 in Omega, we show: (i) if Lambda() = Lambda(-) = mu(1), then (S) has no solution; and (ii) if Lambda(+) ,Lambda(-) is an element of (mu(1), mu(1) + delta), for some delta > 0 small enough, then u(i) < 0 in Omega and partial derivative u(i)/part ial derivative nu > 0 on partial derivative Omega (1 less than or equa l to i less than or equal to n). Our methods for the system (S) are co mpletely different from the case n = 1 (a single equation). For n(n)gr eater than or equal to 2 and p not equal 2, mild additional hypotheses are imposed on the domain Omega and the matrix A =1 (a(ij))(i,j=1).