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Mukhtar Karazym

Publications and source records attributed to Mukhtar Karazym.

6 recordsLinked to original sources
arXiv2026

Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations

We prove the existence of optimizers for a class of homogeneous Gagliardo-Nirenberg inequalities of the form $$ \|D^{s}ϕ\|_{L^q} \leq C \|D^{s_1}ϕ\|_{L^p}^{1-θ} \|D^{s_2}ϕ\|_{L^2}^θ, \quad s_1 s_1$, the problem reduces, after shifting the derivative orders, to the result of Bellazzini, Frank and Visciglia, whereas the case $s\leq s_1$ follows from the present paper. As an application, we study normalized solutions of the associated Euler-Lagrange equations under the constraint $$ \|D^{s_1}ψ\|_{L^p}^p=λ>0. $$ In particular, the case $p=2$ includes the stationary equation arising from the Majda-McLaughlin-Tabak (MMT) model.

math.AP↗
arXiv2026

Traveling Waves for Nonlocal Derivative Nonlinear Schrödinger Equations: A Variational Characterization

We establish several existence results for traveling-wave solutions of the nonlocal derivative nonlinear Schrödinger equation with general coefficients by variational methods. We study associated minimization problems in the subcritical and critical cases and prove the existence of a minimizer in each case. Finally, we derive Pohozaev-type identities and use them to establish corresponding nonexistence results.

math.AP↗
arXiv2025

Subelliptic $p$-Laplacian spectral problem for Hörmander vector fields

Based on variational methods, we study the spectral problem for the subelliptic $p$-Laplacian arising from smooth Hörmander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show Hölder regularity of eigenfunctions. Moreover, we determine the best constant for the $L^{p}$-Poincaré-Friedrichs inequality for Hörmander vector fields as a byproduct.

math.AP↗
arXiv2020

Trace formulae of potentials for degenerate parabolic equations

In this paper, we analyze main properties of the volume and layer potentials as well as the Poisson integral for a multi-dimensional degenerate parabolic equation. As consequences, we obtain trace formulae of the heat volume potential and the Poisson integral which solve Kac's problem for degenerate parabolic equations in cylindrical domains.

math.AP↗
arXiv2020

Multidimensional inverse Cauchy problems for evolution equations

We discuss inverse problems to finding the time-dependent coefficient for the multidimensional Cauchy problems for both strictly hyperbolic equations and polyharmonic heat equations. We also extend our techniques to the general inverse Cauchy problems for evolution equations.

math.AP↗