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In this paper, we investigate the Gibbs measures associated with the focusingnonlinear Schr\"odinger equation with an anharmonic potential. We establish adichotomy for normalizability and non-normalizability of the Gibbs measures inone dimension and higher dimensions with radial data. This extends a recentresult of the third and fourth authors with Robert and Seong (2022), where thefocusing Gibbs measures with a harmonic potential were addressed. Notably, inthe case of a subharmonic potential, we identify a novel critical nonlinearity(below the usual mass-critical exponent) for which the Gibbs measures exhibit aphase transition. The primary challenge emerges from the limited understandingof eigenvalues and eigenfunctions of the Schr\"odinger operator with ananharmonic potential. We overcome the difficulty by employing techniquesrelated to a recent work of the first two authors (2022).

Statistical mechanics of the radial focusing nonlinear Schr\"odinger equation in general traps