On a class of strongly indefinite Schrödinger equations with Stein–Weiss convolution parts and supercritical exponential growth in \(\mathbb {R}^2\)

We study the following class of strongly indefinite Schrödinger equations with Stein–Weiss convolution parts: \(\begin{aligned} -\Delta u+V(x) u =\frac{1}{|x|^\beta }\left( \int _{\mathbb {R}^2}\frac{F(u)}{|x-y|^\mu |y|^\beta }dy\right) f(u),~x\in \mathbb {R}^2, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+V(x)u=\frac{1}{{|x|}^{\beta }}\left({\int }_{{\mathbb{R}}^2},\frac{F(u)}{{|x-y|}^{\mu }{|y|}^{\beta }},d,y\right)f(u),x\in {\mathbb{R}}^2,\end{array}$$ where \(V\in \mathcal {C}^0(\mathbb {R}^2)\) $V\in {\mathcal{C}}^0({\mathbb{R}}^2)$ is bounded, \(\beta >0\) $\beta >0$ , \(0<\mu <2\) $0<\mu <2$ with \(0<2\beta +\mu <2\) $0<2\beta +\mu <2$ and F is the primitive of f that fulfills the supercritical exponential growth in the Trudinger–Moser sense. When 0 belongs to a spectral gap of \(-\Delta +V\) $-\mathrm{\Delta }+V$ , by introducing some new techniques and analytic skills, we exploit the linking-type argument to investigate the existence of nontrivial solutions for the given equation. In particular, the supercritical exponential growth on f would be allowed which should be regarded as one of main contributions in this article. Moreover, we also contemplate the case that the nonlinearity f possesses the critical exponential growth at infinity.