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

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.