A nonhomogeneous Schrödinger equation involving nonlinearity with exponential critical growth and potential which can vanish at infinity

The main purpose of this work is to study the existence of solutions for the following class of nonhomogeneous Schrödinger equations −Δu+V(x)u=f(u)+h(x),x∈R2,where V is a continuous potential which can vanish at infinity, the nonlinearity f possesses maximal growth range and h belong to the dual of a functional space. Using Ekeland variational principle and the Mountain Pass theorem, we prove that the above problem has at least two nontrivial weak solutions provided that h is sufficiently small