On the planar Choquard equation with indefinite potential and critical exponential growth

Abstract

In the present paper, we study the following planar Choquard equation:{−Δu+V(x)u=(Iα⁎F(u))f(u),x∈R2,u∈H1(R2), where V(x) is an 1-periodic function, Iα:R2→R is the Riesz potential and f(t) behaves like ±eβt2 as t→±∞. A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator −△+V. In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) [16], of Alves-Cassani-Tarsi-Yang (2016) [4], of Ackermann (2004) [1], and of Alves-Germano (2018) [5].