Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R-2, -Delta V(x)u = (1/vertical bar x vertical bar(mu) * F(u)/ vertical bar x vertical bar(beta)) f(u)/vertical bar x vertical bar(beta) + epsilon h(x) in R-2, where V is a positive continuous potential, 0 < mu < 2, beta > 0, 2 beta, mu <= 2, epsilon is a small parameter and F(s) is the primitive function of f(8). Suppose that the nonlinearity f (s) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods. (C) 2019 Elsevier Inc. All rights reserved.