The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains

In this paper, we study the existence and concentration of weak solutions to the p-Laplacian type elliptic problem {-epsilon Delta(p)u + V(z)vertical bar u vertical bar(p) (2)u - f(u) = 0 in Omega, u = 0 on partial derivative Omega, u > 0 in Omega, N > p > 2, where Q is a domain in R-N, possibly unbounded, with empty or smooth boundary, epsilon is a small positive parameter, f is an element of C-1 (R+, R) is of subcritical and V : R-N -> R is a locally Holder continuous function which is bounded from below, away from zero, such that inf(A) V < min(partial derivative A) V for some open bounded subset A of Omega. We prove that there is an epsilon(0) > 0 such that for any epsilon is an element of(0, epsilon 0], the above mentioned problem possesses a weak solution u(epsilon) with exponential decay. Moreover, u(epsilon) concentrates around a minimum point of the potential V in A. Our result generalizes a similar result by del Pino and Felmer (1996) for semilinear elliptic equations to the p-Laplacian type problem.