<p>We study the stationary Swift–Hohenberg equation <Equation ID="Equ22"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3010_Article_Equ22.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="226" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} (\Delta + 1)^2 u - \alpha u - \beta u^2 + u^3=0 \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mi>u</mi> <mo>-</mo> <mi>α</mi> <mi>u</mi> <mo>-</mo> <mi>β</mi> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>u</mi> <mn>3</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in the whole space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3010_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {R}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3010_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(2 \le n \le 7\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>. We develop and modify the variational approach introduced by Lerman et al. (Nonlinear Anal TMA 190:1–21, 2020) and obtain a series of periodic solutions with certain additional symmetries.</p>