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

Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -\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 ∈ L 2 / ( 2 − q ) ( R 2 ) 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 x x , 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.