Normalized solutions to a class of Kirchhoff type equations with a logarithmic perturbation

TL;DR

This paper investigates normalized solutions to a Kirchhoff type equation with a logarithmic perturbation, developing a variational framework to establish existence, multiplicity, and asymptotic behavior of solutions across different parameter regimes.

Contribution

It introduces a unified variational approach using Orlicz spaces and Pohozaev constraints to find multiple solutions, including ground states and solutions with specific asymptotic properties.

Findings

Existence of positive radial ground states for certain p ranges.

Presence of two solutions for small mass when p > 14/3.

Asymptotic behavior of solutions as mass approaches zero.

Abstract

This paper is devoted to the study of normalized solutions to the Kirchhoff type equation with a logarithmic perturbation\[-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 \,\mathrm{d}x \right) \Delta u=\lambda u+|u|^{p-2}u+u\log u^2,\quad x \in\mathbb{R}^3, \]under the normalized constraint $\int_{\mathbb{R}^3} u^2 \,\mathrm{d}x = c^2$, where $a,b>0$, $2<p\leq 6$, $c>0$ is a constant, and $\lambda\in\mathbb{R}$ emerges as a Lagrange multiplier which is not a priori known. A unified variational framework is developed based on Orlicz spaces together with the Pohozaev constraint method and refined fiber map analysis. For $2<p<\frac{14}{3}$ or $p=\frac{14}{3}$ with small mass, the energy functional is bounded from below and admits a positive radial ground state minimizer. For $\frac{14}{3}<p<6$, where the energy functional is unbounded from below, we establish the existence of two normalized solutions for small mass: a ground state $u_c^+$ obtained via local minimization, and a second solution $u_c^-$ obtained via minimization on the negative component of the Pohozaev manifold. For the Sobolev critical case $p=6$, we construct a ground state solution and, under a technical condition on the parameters, a second solution by introducing a proper auxiliary functional and precise energy estimates with Aubin-Talenti bubbles. Asymptotically as $c\to0^+$, the $L^{2}$ norm of the gradient of ground state solution vanishes for $2<p\le6$. Surprisingly, for $\frac{14}{3}<p<6$, the $L^{2}$ norm of the gradient of the second solution diverges to infinity as $c\to 0^+$, while for $p=6$ it concentrates around the Aubin-Talenti bubble with energy converging to the energy level of the corresponding critical Kirchhoff equation.