Schrödinger equations in [formula omitted] with critical exponential growth and concave nonlinearities

Abstract

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.