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In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality: $$\int_\mathbb{R}|u|^{\alpha+2}\,dx\leq C_{\alpha,s}\left(\int_\mathbb{R}|(-\Delta)^{s/2}u|^2\,dx\right)^{\alpha/4s}\left(\int_\mathbb{R}|u|^2\,dx\right)^{\alpha(2s-1)/4s+1},$$ for some admissible $\alpha$. I think the relativistic generalization of this inequality should hold as next form: $$\int_\mathbb{R}|u|^{\alpha+2}\,dx\leq C_{\alpha,s,m}\left(\int_\mathbb{R}|(-\Delta+m^2)^{s/2}u|^2\,dx\right)^{\alpha/4s}\left(\int_\mathbb{R}|u|^2\,dx\right)^{\alpha(2s-1)/4s+1},$$ with an optimizer in $H^s(\mathbb{R})$. But I am afraid that I cannot find a suitable reference. So I wonder if the existence of this optimizer proven already, or reasonable enough to deduce easily somehow, otherwise I can try the same approach of M. Weinstein.

Thank you in advance.

References

[1] Frank, Rupert L., and Enno Lenzmann. "Uniqueness of non-linear ground states for fractional Laplacians in R." Acta mathematica 210.2 (2013): 261-318.

[2] Weinstein, Michael I. "Modulational stability of ground states of nonlinear Schrödinger equations." SIAM journal on mathematical analysis 16.3 (1985): 472-491.

[3] Weinstein, Michael I. "Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation." Communication in partial Differential Equation 12.10 (1987): 1133-1173.

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    $\begingroup$ Is your question about the existence of the optimizer, or is your question about the validity of the inequality itself? For the inequality: if you use the obvious definition $(-\Delta + m^2)^{s/2} u = \mathcal{F}^{-1}( (|\xi|^2 + m^2)^{s/2} \hat{u})$ then the inequality holds because $$ \| (-\Delta)^{s/2} u\|_{L^2} \leq \|(-\Delta + m^2)^{s/2} u\|_{L^2} $$ $\endgroup$
    – Willie Wong
    Aug 22, 2022 at 21:12
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    $\begingroup$ The fact that the optimizer exists should be by now an application of Concentration-Compactness type arguments. Standard treatments usually work with the case $s = 1$, but the "concentration compactness" part of the argument is generally independent of the "sobolev inequality" part, so modifying the arguments found in, e.g. Tintarev's Concentration compactness textbook shouldn't be too difficult. $\endgroup$
    – Willie Wong
    Aug 22, 2022 at 21:22
  • $\begingroup$ @WillieWong Sorry, my question was indeed vague. I was more interested in the existence of the optimizer. Thank you so much for the reference, I will check it out. Have a great day. $\endgroup$
    – Jingeon An-Lacroix
    Aug 23, 2022 at 11:50
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    $\begingroup$ As a side remark: since you wrote down the scale invariant version of the inequality, for concentration compactness you need to account for both translation in $\mathbb{R}$ and for scaling. Even if you take the main concentration compactness method as a black box, filling in all the details would still take at least 1 entire afternoon and 5 or more pages. $\endgroup$
    – Willie Wong
    Aug 23, 2022 at 13:22
  • $\begingroup$ @WillieWong That would be awesome if I can do it in one afternoon :) Thank you for your help! $\endgroup$
    – Jingeon An-Lacroix
    Aug 23, 2022 at 14:13

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