Choquard type equations with asymptotically linear nonlinearities

In this work, we consider the following class nonlocal elliptic problems: \(\begin{aligned} \left\{ \begin{array}{lr} -\Delta u + V(x) u = [I_\alpha * F(x,u)] f(x,u), & x\in \mathbb {R}^N,\\ u \in H^1(\mathbb {R}^N), & \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+V(x)u=[I_{\alpha }\ast F(x,u)]f(x,u),\hfill & \hfill x\in {\mathbb{R}}^N,\\ u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(N \ge 3\) $N\ge 3$ , \(I_\alpha \) $I_{\alpha }$ denotes the Riesz potential with \(\alpha \in (0, N)\) $\alpha \in (0,N)$ , \(V: \mathbb {R}^N \rightarrow \mathbb {R}\) $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive continuous potential and the nonlinearity \(f:\mathbb {R}^N\times \mathbb {R}\rightarrow \mathbb {R}\) $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is asymptotically linear at infinity and at the origin in a suitable sense. The nonlocal term \(I_\alpha * F\) $I_{\alpha }\ast F$ is the convolution de \(I_\alpha \) $I_{\alpha }$ with the primitive F(x, u) of f(x, u). Our main results rely on the fact that nonlocal semilinear elliptic problems have nontrivial solutions whenever a kind of crossing of eigenvalues is allowed. Here, we consider an eigenvalue elliptic problem with a nonlocal term driven by the Choquard equation.