Solutions of biharmonic equations with steep potential wells

Abstract

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: ∆2u + [λV (x) − δ]u = |u|p−2u, x ∈ RN, (0.1) where (Formula presented), λ > 0 is a parameter, V (x) is a nonnegative potential function with nonempty interior part of the zero set intV −1(0), 0 < δ < µ0 and µ0 is the principal eigenvalue of ∆2 in the zero set intV −1(0) of V (x). We prove that the equation (0.1) admits a least energy solution which is trapped near the zero set V −1(0) for λ > 0 large enough.