On Choquard problems in \(\mathbb {R}^N\) influenced by the negative part of the spectrum

This article focuses on studying the existence of solutions for the following indefinite Choquard equation in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ : where \(N \ge 2\) $N\ge 2$ . Here, \(\alpha \) $\alpha$ is a positive constant satisfying \(0<\alpha <N\) $0<\alpha <N$ , and \(I_{\alpha }\) $I_{\alpha }$ represents the Riesz potential of order \(\alpha \) $\alpha$ . The potential V(x) is nonperiodic and changes sign, leading to \(\inf \sigma (-\Delta + V)<0\) $inf\sigma (-\mathrm{\Delta }+V)<0$ , where \(\sigma (-\Delta + V)\) $\sigma (-\mathrm{\Delta }+V)$ denotes the spectrum of the operator \(-\Delta +V\) $-\mathrm{\Delta }+V$ , which the problem become indefinite. The exponent p is chosen from the range given by 0.1 \(\begin{aligned} \frac{N-2}{N+\alpha }<\frac{1}{p}<\dfrac{1}{2}<\frac{N}{N+\alpha }. \end{aligned}\) $$\begin{array}{c}\hfill \frac{N-2}{N+\alpha }<\frac{1}{p}<{\displaystyle \frac{1}{2}}<\frac{N}{N+\alpha }.\end{array}$$ In this work, we establish the existence of nontrivial solutions by employing significant outcomes from spectral theory, along with a version of the linking theorem presented by [13], and the interaction between translated solutions of the problem at infinity.