Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth

We study a class of gauged nonlinear Schrödinger equations− Δ u + ω u + λ∫ | x | ∞h ( s ) s u 2 ( s ) d s +h 2 ( | x | ) | x | 2) u = f ( u ) inR 2 ,u ∈ H r 1 ( R 2 ) ,where ω , λ >0 and h ( s ) = ∫ 0 sr 2 u 2 ( r ) d r . Under some suitable assumptions on f with critical exponential growth, we obtain a positive ground state solution by the non‐Nehari manifold method. When f ( u ) has subcritical exponential growth, we prove the existence of a positive ground state solution by using a new approach. Our results generalize and improve the ones in Ji‐Fang [J. Math. Anal. Appl. 450 (2017) 578‐591], Byeon et al [J. Funct. Anal. 263 (2012) 1575‐1608], and some other related literatures.