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 {fx1-1} where Ω is a domain in ℝN, possibly unbounded, with empty or smooth boundary, ɛ is a small positive parameter, f ∈ C1(ℝ+, ℝ) is of subcritical and V: ℝN → ℝ is a locally Hölder continuous function which is bounded from below, away from zero, such that infΛV < min∂ΛV for some open bounded subset Λ of Ω. We prove that there is an ɛ0 > 0 such that for any ɛ ∈ (0, ɛ0], the above mentioned problem possesses a weak solution uε with exponential decay. Moreover, uε concentrates around a minimum point of the potential V in Λ. Our result generalizes a similar result by del Pino and Felmer (1996) for semilinear elliptic equations to the p-Laplacian type problem.