multiple existence of solutions for a nonhomogeneous elliptic problem on rn

let n ≥ 3, 2^∗ = 2n/(n − 2) and p ∈ (2, 2^∗). our purpose in this paper is to consider the multiple existence of solutions of problem −ε²δu + u = |u|^p−2u + λf u ∈ h¹(rⁿ ), where ε, λ > 0, f ∈ l²(rⁿ ) ∩ c²(rⁿ ), f ≥ 0 and f ̸≡ 0. we will show the effect of the shape of f to the number of solutions.