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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},whereλ&gt;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), andf(x,t)f\left(x,t)are 1-periodic with respect toxx, andf(x,t)f\left(x,t)has critical exponential growth att=∞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&gt;0\lambda \gt 0) is very helpful to the existence of nontrivial solutions for equation ( 0.1 ) in one sense.

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