A remark on the existence and multiplicity result for a nonlinear elliptic problem involving the p-Laplacian

In this work, motivated by Wu (J Math Anal Appl 318: 253-270, 2006), and using recent ideas from Brown and Wu ( J Math Anal Appl 337: 1326-1336, 2008), we prove the existence of nontrivial nonnegative solutions to the following nonlinear elliptic problem: {-Delta(p)u + m(x) u(p-1) = lambda a(x) u(alpha-1) + b(x) u(beta-1), x is an element of Omega, u = 0, x is an element of partial derivative Omega, Here Delta(p) denotes the p-Laplacian operator defined by Delta(p)z = div (vertical bar del z vertical bar(p-2)del z), p > 2, Omega subset of R-N is a bounded domain with smooth boundary, 2 < beta < p < alpha < p* (p* = pN/N-p if N > p, p* = infinity if N <= p), lambda is an element of R \ {0}, the weight m(x) is a bounded function with parallel to m parallel to(infinity) > 0 and a(x), b(x) are continuous functions which change sign in (Omega) over bar.