Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth

We consider a Kirchhoff type elliptic problem; {-(1 + alpha integral(Omega) vertical bar del u vertical bar(2)dx) Delta u = f(x, u), u >= 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R-2 is a bounded domain with a smooth boundary partial derivative Omega, alpha > 0 and f is a continuous function in (Omega) over bar x R. Moreover, we assume f has the Trudinger-Moser growth. We prove the existence of solutions of (P), so extending a former result by de Figueiredo-Miyagaki-Ruf [11] for the case alpha = 0 to the case alpha > 0. We emphasize that we also show a new multiplicity result induced by the nonlocal dependence. In order to prove this, we carefully discuss the geometry of the associated energy functional and the concentration compactness analysis for the critical case