A MULTIPLICITY RESULTS TO A p - q LAPLACIAN SYSTEM WITH A CONCAVE AND SINGULAR NONLINEARITIES

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. - Delta pu-Delta qu= lambda f(x)|u|r-2u +nu 1 -alpha 2 -alpha - beta h(x)|u|-alpha|v|1-beta in omega, - Delta pv-Delta qv= mu g(x)|v|r-2v +nu 1 -beta 2 -alpha -beta h(x)|u|1-alpha|v|-beta in omega, u, v > 0 in omega, u = v = 0 on partial differential omega where 0 < alpha < 1, 0< beta <1, 2- alpha -beta < q < N(p-1) N-p < p < r < p*, with p* = NpN-p. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least cat(omega) + 1 number of solutions.