POLYHARMONIC SYSTEMS INVOLVING CRITICAL NONLINEARITIES WITH SIGN-CHANGING WEIGHT FUNCTIONS

This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions (-Delta)(m)u = lambda f(x)vertical bar u vertical bar(r-2) u + beta/beta + gamma h(x)vertical bar u vertical bar(beta-2)u vertical bar v vertical bar(gamma) in Omega, (-Delta)(m)v = mu g(x)vertical bar v vertical bar(r-2) v + gamma/beta + gamma h(x)vertical bar u vertical bar(beta)vertical bar v vertical bar(gamma-2)v in Omega, D(k)u = D(k)v = 0 for all vertical bar k vertical bar <= m - 1 on partial derivative Omega, where (-Delta)(m) denotes the polyharmonic operators, Omega is a bounded domain in R-N with smooth boundary partial derivative Omega, m is an element of N, N >= 2m + 1, 1 < r < 2 and beta > 1, gamma > 1 satisfying 2 < beta + gamma <= 2(m)* with 2(m)* = 2N/N-2m as a critical Sobolev exponent and A, mu > O. The functions f , g and h : (Omega) over bar -> R are sign-changing weight functions satisfying f, g is an element of L-alpha(Omega) and h is an element of L-infinity(Omega) respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter (lambda, mu) is an element of R-+(2) \ {(0, 0)}.