<p>We study the elliptic Gross–Pitaevskii system with both linear and nonlinear coupling on expanding spherical shells with Neumann boundary conditions. We derive exact asymptotic formulas for the Morse index of radial <i>n</i>-mode solutions as the radius of the spherical shells approaches infinity. We explore how the formula depends on system parameters such as the diffusion parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>, the linear coupling parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>, the shell radius <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(R\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>R</mi> </math></EquationSource> </InlineEquation>, and the multilayer parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>. Our results provide a deeper understanding of the behavior of these solutions in the limit of large spatial domains, contributing to the broader analysis of the Gross–Pitaevskii system in high-dimensional settings.</p>