Existence and behavior of positive solutions for a class of quasilinear elliptic problems with discontinuous nonlinearity

In this paper, we study the existence and behavior of positive solutions of the following quasilinear elliptic problems with discontinuous nonlinearities: {-Delta u + V(x)u - kappa u Delta(u(2)) = H(u - delta)f(x, u), in R-N, (P-delta) u is an element of D-1,D-2(R-N) boolean AND (WlocRN)-R-2,2(), where delta, kappa > 0, N >= 3, V: R-N -> R is a nonnegative continuous function, which can vanish at infinity, that is, V(x) -> 0 as vertical bar x vertical bar -> infinity, f : R-N x R -> R is a Caratheodory function and H is the Heaviside function. Via a suitable nonsmooth truncation, we apply the penalization method combined with the Mountain Pass Theorem for locally Lipschitz functional to obtain a positive solution u delta of (P-delta) for all delta > 0. Besides, we establish the convergent behavior of positive solution sequence {u(delta)}, that is, u(delta) -> u(0) in D-1,D-2(R-N) as delta -> 0(+), where u(0) is a positive solution of (P-0).