A nonhomogeneous Schrodinger 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 Schrodinger equations - increment u + V(x)u = f (u) + h(x), x is an element of 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.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).