On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1)−Δu+V(x)u=f(x,u)+λa(x)∣u∣q−2u,x∈R2,-\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where λ>0\lambda \gt 0, q∈(1,2)q\in \left(1,2), a∈L2/(2−q)(R2)a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2}), V(x)V\left(x), and f(x,t)f\left(x,t) are 1-periodic with respect to xx, and f(x,t)f\left(x,t) has critical exponential growth at t=∞t=\infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0\lambda \gt 0) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.