UNIQUENESS OF POSITIVE SOLUTIONS FOR NONLINEAR COOPERATIVE SYSTEMS WITH THE P-LAPLACIAN

In this article we obtain the existence and uniqueness of a positive s olution to the following strictly cooperative elliptic system, [GRAPHI CS] Here, Omega is a bounded domain in R(N) whose boundary partial der ivative Omega is of class C-2,C-alpha for some alpha is an element of (0,1), Delta(p) denotes the p-Laplacian defined by Delta(p)u def over equals sign div (\del u\(p-2)del u) p is an element of (1,infinity), a nd the coefficients a(ij) (1 less than or equal to i,j less than or eq ual to n) are assumed to be constants satisfying a(ij) > 0 for i not e qual j (a strictly cooperative system) and a(ii) < 0. We assume that t he functions f(i)(x,u(1),...,u(n)) satisfy f(i) is an element of C-alp ha(<(Omega)over bar> x R(+)(n)), partial derivative f(i)/partial deriv ative u(j) is an element of C(Omega x (0,infinity)(n)) and f(i) greate r than or equal to 0, partial derivative f(i)/partial derivative u(j) greater than or equal to 0 (1 less than or equal to i,j less than or e qual to n), and each lambda ''bar arrow pointing right'' lambda(1-p) f (i)(x,lambda u(1),...,lambda u(n)) is a strictly monotone decreasing f unction lambda is an element of (0,infinity). Our methods combine weak and strong comparison principles with standard tools for strongly sub -homogeneous monotone mappings.