EXISTENCE OF POSITIVE SOLUTION FOR KIRCHHOFF TYPE PROBLEM WITH CRITICAL DISCONTINUOUS NONLINEARITY

In this paper we are concerned with existence of positive solution to the class of nonlinear problems of the Kirchhoff type given by L-epsilon (u) = H (u - beta) f (u) + u(2*-1 )in R-N, u is an element of H-1 (R-N) boolean AND W-2,W- q/q-1 (RN),W- where N >= 3, q is an element of (2, 2*), epsilon, beta > 0 are positive parameters, f : R -> R is a continuous function, H is the Heaviside function, i.e., H(t) = 0 if t <= 0, H(t) = 1 if t > 0 and L-epsilon(u) := [ M (1/epsilon(N-2) integral(RN) vertical bar del u vertical bar(2 )dx + 1/epsilon(N) integral(RN) V(x)vertical bar u vertical bar(2) dx)] [-epsilon(2 )Delta u + V (x) u]. The function M is a general continuous function. The function V is a positive potential that satisfies following hypothesis: or V satisfies the Palais-Smale condition or there is a bounded domain Omega in R-N such that V has no critical point in partial derivative Omega. Here we use a suitable truncation to apply a version of the penalization method of Del Pino and Felmer [16] combined with the Mountain Pass Theorem for locally Lipschitz functional.