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

Abstract

This article focuses on studying the existence of solutions for the following indefinite Choquard equation in $\mathbb {R}^{N}$:

where $N \ge 2$. Here, $\alpha$ is a positive constant satisfying $0<\alpha <N$, and $I_{\alpha }$ represents the Riesz potential of order $\alpha$. The potential V(x) is nonperiodic and changes sign, leading to $\inf \sigma (-\Delta + V)<0$, where $\sigma (-\Delta + V)$ denotes the spectrum of the operator $-\Delta +V$, which the problem become indefinite. The exponent p is chosen from the range given by

$$\begin{aligned} \frac{N-2}{N+\alpha }<\frac{1}{p}<\dfrac{1}{2}<\frac{N}{N+\alpha }. \end{aligned}$$

(0.1)

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.

Appendix A

Let $\overline{B_R}$ be the closure of the ball of radius $R>0$ in a finite-dimensional space $E^-$, that is generated by the functions $\phi _1,\ldots ,\phi _k$. We would like to remind that when we write $\Upsilon (1)$ or $\Upsilon ^+(1)$ in the following integrals, the variable in question is implied. Additionally, both functions are functions that depend on $z\in \mathbb {R}^N$. We want to prove that

$$\begin{aligned} \displaystyle \lim _{|z|\rightarrow \infty } \left[ \,\,\,\int \limits _{\mathbb {R}^{N}}\left( I_{\alpha } *|\Upsilon ^{+}(1)+|v^-||^p\right) |\Upsilon ^{+}(1)+|v^-||^{p} \textrm{d}x - \int \limits _{\mathbb {R}^{N}}\left( I_{\alpha } *|\Upsilon (1)+|v^-||^p\right) |\Upsilon (1)+|v^-||^{p} \textrm{d}x\right] = 0,\nonumber \\ \end{aligned}$$

(5.1)

uniformly on $v^-\in \overline{B_R}$. Indeed, we have proved in (4.7) and (4.8) that

$$\begin{aligned} \mathcal {J}_z:= & {} \dfrac{1}{2p} \displaystyle \int \limits _{\mathbb {R}^N}\displaystyle \int \limits _{\mathbb {R}^N}\dfrac{1}{|x-y|^{N-\alpha }} \left( \dfrac{}{}|\Upsilon _y(t_z)|^p|\Upsilon _x(t_z)|^p - |\Upsilon ^+_y(t_z)|^{p}|\Upsilon ^+_x(t_z)|^{p}\right) \textrm{d}x\textrm{d}y \\ {}\le & {} \dfrac{C}{2p} \displaystyle \int \limits _{\mathbb {R}^N}\displaystyle \int \limits _{\mathbb {R}^N}\dfrac{1}{|x-y|^{N-\alpha }}\left( |\Upsilon ^-_y(t_z)|+|\Upsilon ^-_x(t_z)|\right) \left( \dfrac{}{} |\Upsilon _x(t_z)|^{p-1}|\Upsilon _y(t_z)|^p + |\Upsilon ^-_x(t_z)|^{p-1}|\Upsilon _y(t_z)|^p \right. \\{} & {} \left. + \ |\Upsilon _x(t_z)|^{p-1} |\Upsilon ^-_y(t_z)|^{p} + |\Upsilon ^-_x(t_z)|^{p-1}|\Upsilon ^-_y(t_z)|^p +|\Upsilon _x(t_z)|^{p}|\Upsilon _y(t_z)|^{p-1} + |\Upsilon ^-_x(t_z)|^{p}|\Upsilon _y(t_z)|^{p-1}\right. \\{} & {} \left. + \ |\Upsilon _x(t_z)|^{p} |\Upsilon ^-_y(t_z)|^{p-1} + |\Upsilon ^-_x(t_z)|^{p}|\Upsilon ^-_y(t_z)|^{p-1} \dfrac{}{}\right) \textrm{d}x\textrm{d}y \\\le & {} C \displaystyle \sum _{i=1}^{16} e^{-\mu _{i,z}|z|}, \ \text {for all} \ z\in \mathbb {R}^N. \end{aligned}$$

Therefore, to prove the limit (5.1), we proceed in the same way to obtain several integrals that depend on $|v^-|$. Since $v^-\in \overline{B_R}$, we just use the exponential decay of the functions in $E^-$, which is guaranteed by the finite dimension of $E^-$. Therefore, the limit (5.1) is uniformly on $v^-\in \overline{B_R}$, as we wished to prove.