Abstract
In this work, we are concerned with the existence and nonexistence of large positive weak solutions for the following class of weighted (p, q) Laplacian nonlinear system
-(Delta(P,p) + rho(p)Lambda(p))u = lambda a(x)f(v) in Omega,
-(Delta(Q,q) + rho(q)Lambda(q))v = mu b(x)g(u) in Omega,
u = v = 0 on partial derivative Omega, }
where Delta(P,p) with p > 1 and P = P (x) is a weight function, denotes the weighted p-Laplacian defined by Delta(P,pu) equivalent to div[P(x) vertical bar del u vertical bar(p) (2)del u], and Lambda(p) (u) = vertical bar u vertical bar(p-2) u, rho(p), rho(q), lambda, mu are positive constants, a, b are weight functions and Omega subset of R-N is a bounded domain with smooth boundary partial derivative Omega. We prove the existence of a large positive weak solutions for lambda, mu large when lim(x ->+infinity) f1/p-1(M(g(x))1/q-1)/x = 0, for every M > 0. In our approach, we do not assume any sign-changing conditions on a or b: The proof of the main results is based on the sub-supersolutions method. Application and concluding remark are provided to demonstrate the effectiveness of our results.