Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: {-a(integral(Omega) vertical bar del u vertical bar(N) dx) Delta U-N = f(X, u)/vertical bar X vertical bar(b) + lambda h(x) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N(N >= 2) and 0 <= b < N. Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: -a(integral(RN) vertical bar del u vertical bar(N) + V(x)vertical bar u vertical bar(N) dx) (Delta(N)u + V(x)vertical bar u vertical bar(N-2) u) =g(x, u)/vertical bar x vertical bar(b) + lambda h(x) in R-N, where N >= 2 and 0 <= b < N. We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser-Trudinger inequality.