A Choquard equation with an almost periodic term

Abstract We consider the existence of nontrivial solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $u\in {H}^{1}\left({\mathbb{R}}^{N}\right)$ for the following nonlinear Choquard equation with an almost periodic term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mo>∗</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>F</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mi>α</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mtext> </m:mtext> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:math> $$-{\Delta}u+u=\left({I}_{\mu }\ast \left(\alpha \left(y\right)F\left(u\right)\right)\right)\alpha \left(x\right)f\left(u\right)\quad \text{in}\quad {\mathbb{R}}^{N},$$ where N ≥ 3, μ ∈ (0, N ), I μ is the Riesz potential, F is the primitive function of f , and α is almost periodic. As one will see, almost periodic functions of several variables lack some important features of periodic functions and almost periodic functions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">R</m:mi> </m:math> $\mathbb{R}$ . This makes our problem more intriguing and challenging.