Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations

In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, − Δ u + u = | u | p s g n ( u ) + f in R N , N ⩾ 3 , 0 < p < 1 , f ∈ L 2 ( R N ) , f > 0 a.e. in R N . We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H 1 ( R N ) ∩ L p + 1 ( R N ) . We study its continuity in the perturbation parameter f at 0.