The higher-dimensional GPEVP ground-state $L^\infty$ bound

Let $d>1$, and let $u_{\text{\tiny\rm gs}}$ denote the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP). Higher-dimensional ground-state bound. The ground state satisfies

$$\|u_{\text{\tiny\rm gs}}\|_{L^\infty} \lesssim \varepsilon^{-d/2}.$$

This bound would extend the strategy used for the one-dimensional localization result to higher dimensions, where the Sobolev embedding and the secant estimate used in the paper are not available in their present form. The conjecture is presented without a resolution in the source.