Existence theory for non-integer logarithmic orders

Determine whether results analogous to those established for the higher-order logarithmic Schrödinger equation with integer odd order m remain valid when the logarithmic order is non-integer.

Background

The paper studies a higher-order logarithmic Schrödinger equation whose nonlinear term contains u(ln|u|)m, under the structural assumption that m is an odd positive integer. This restriction is used in the analysis, including integration-by-parts arguments and sign properties of the nonlinearities.

The authors explicitly identify the extension to non-integer orders as unresolved. Such an extension would require determining whether the existence, compactness, profile-decomposition, and min–max arguments remain valid beyond the integer-order setting.

References

Whether similar results hold for non-integer orders remains an interesting problem.

— Existence of a nonnegative bound state for a higher-order logarithmic Schrödinger equation  (2609.20096 - Ou et al., 17 Sep 2026) in Remark 1.2(vi), Introduction

Therefore, under the same condition, the estimates hold for every real $m\geq1$. Consequently, we conjecture that this method of taking the positive part may eliminate the restriction on $m$ being an odd integer when estimating the growth of a function, and may be of practical value for generalizing the current high-order perturbation to a more general exponent $m\geq 1$.

— Existence of a nonnegative bound state for a higher-order logarithmic Schrödinger equation  (2609.20096 - Ou et al., 17 Sep 2026) in Remark 2.4, subsection “Uniform estimates for approximation”