Ground states for planar axially Schrödinger–Newton system with an exponential critical growth

Abstract

Abstract In this paper, we study the following planar Schrödinger–Newton system: { − Δ u + V ( x ) u + λ ϕ u = f ( x , u ) in R 2 , Δ ϕ = u 2 in R 2 , $$ \left \{ \textstyle\begin{array}{l} -\Delta u+ V(x)u+\lambda\phi u= f(x,u)\quad \textrm{in } \mathbb{R}^{2},\\ \Delta\phi=u^{2}\quad\textrm{in } \mathbb{R}^{2}, \end{array}\displaystyle \right . $$ where V, f are axially symmetric about x, V is positive, and f is super-linear at zero and exponential critical at infinity. Using a weaker condition [ f ( x , u ) u 3 − f ( x , t u ) ( t u ) 3 ] sign ( 1 − t ) + θ V ( x ) | 1 − t 2 | ( t u ) 2 ≥ 0 , ∀ x ∈ R 2 , t > 0 , u ≠ 0 $$ \biggl[\frac{f(x,u)}{u^{3}}-\frac{f(x,tu)}{(tu)^{3}} \biggr]\operatorname {sign}(1-t)+ \theta V(x)\frac{ \vert 1-t^{2} \vert }{(tu)^{2}}\geq0,\quad \forall x\in \mathbb{R}^{2}, t>0, u\neq0 $$ with θ ∈ [ 0 , 1 ) $\theta\in[0,1)$ instead of the Nehari type monotonic condition on f ( x , u ) | u | 3 $\frac{f(x,u)}{|u|^{3}}$ , we obtain a ground state solution of the above problem via variational methods.

Keywords

• Schrödinger–Newton system
• Axial symmetry
• The exponential critical growth
• Ground states