Ground State Solution for Strongly Indefinite X-Ray Free Electron Laser Schrödinger Equation

In this paper, we are concerned with the following X-ray free electron laser Schrödinger equation \(\begin{aligned} (i\nabla -A(x))^2u+V(x)u-\frac{\mu }{|x|}u=g(x,|u|)u,\ x\in \mathbb {R}^N,\ N\ge 2, \end{aligned}\) $$\begin{array}{c}\hfill {(i\mathrm{\nabla }-A(x))}^{2}u+V(x)u-\frac{\mu }{|x|}u=g(x,|u|)u,x\in {\mathbb{R}}^N,N\ge 2,\end{array}$$ where V is the electric potential, A is the magnetic potential and \(\mu \ge 0\) $\mu \ge 0$ is a constant. We mainly study the existence/nonexistence of ground states and their asymptotic behavior properties in two cases: One is that \(N=2\) $N=2$ with subcritical exponential growth and critical exponential growth. The other is that \(N\ge 3\) $N\ge 3$ under the indefinite case with two types of polynomial growth: super-quadratic and asymptotically quadratic. By using subtle estimates and establishing a new energy estimate inequality in complex field, we overcome the difficulty arising in strongly indefinite structure associated with the energy functional and the presence of magnetic field. Our results extend and complement the present ones in the literature.