Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents

In this paper, we consider the equation -Delta(p)u = lambda vertical bar u vertical bar(p*-1)u + f(x, u) in R-N, with discontinuous nonlinearity, where 1 < p < N, lambda > 0 is a real parameter and p* = N-p/N-p is the critical Sobolev exponent. Under proper conditions on f, applying the nonsmooth critical point theory for locally Lipschitz functionals, we obtain at least one nontrivial nonnegative solution provided that lambda < lambda(0) and for any k is an element of N. it has k pairs of nontrivial solutions if lambda < lambda(k), where lambda(0) and lambda(k) are positive numbers. In particular, we obtain the existence results for f is discontinuous in just one point. (C) 2011 Elsevier Inc. All rights reserved.