A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain

In this paper, our main purpose is to establish the existence of multiple solutions of a class of p-q-Laplacian equation involving concave-convex nonlinearities: {- Delta(p)u - Delta(q)u = theta V (x)vertical bar u vertical bar(r-2)u + vertical bar u vertical bar(P*-2)u + lambda f (x, u), x epsilon Omega, u = 0, x epsilon partial derivative Omega where Omega is a bounded domain in R-N, lambda, theta > 0, 1 < r < q < p < N and p* = Np/N-p is the N-p critical Sobolev exponent, Delta(s)u = div(vertical bar del(u)vertical bar(s-2)del u) is the s-Laplacian of u. We prove that for any lambda epsilon (0, lambda*), lambda* > 0 is a constant, there is a theta* > 0, such that for every 0 E (0, 0*), the above problem possesses infinitely many weak solutions. We also obtain some results for the case 1 < q < p < r < p*. The existence results of solutions are obtained by variational methods. (C) 2011 Elsevier Inc. All rights reserved.