Existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions

In this paper, the existence and multiplicity results of solutions for the following quasilinear elliptic problem {-div(vertical bar x vertical bar(-ap)vertical bar del u vertical bar(p-2)del u) + h(x)vertical bar u vertical bar(p-2)u = g(x)vertical bar u vertical bar(r-2)u, x is an element of Omega, vertical bar x vertical bar(-ap)vertical bar del u vertical bar(p-2) partial derivative u/partial derivative v = lambda f(x)vertical bar u vertical bar(q-2)u, x is an element of partial derivative Omega (0.1) are established, where Omega is an exterior domain in R-N with the compact and smooth boundary partial derivative Omega. By the variational methods, we prove that the problem (0.1) has at least two positive solutions. At the last part of the paper, we also consider the critical case and prove the existence of solution by concentration-compactness principle. (C) 2013 Elsevier Inc. All rights reserved.