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In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical L ˆ r space where L ˆ r = { f ∈ S ′ ( R ) | ‖ f ‖ L ˆ r = ‖ f ˆ ‖ L r ′ < ∞ } . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to L ˆ r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]