Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four

TL;DR

This work analyzes a critical Kirchhoff-type elliptic equation with a logarithmic perturbation in dimension four, incorporating a nonlocal term $\bigl(1+b\int_\Omega|\nabla u|^2 dx\bigr)$. The authors deploy $Ekeland$'s variational principle, Brézis-Lieb splitting, and careful handling of the logarithmic nonlinearity to prove the existence of a local minimum and a least-energy weak solution in $H_0^1(\Omega)$ under structured parameter regimes. They show the nonlocal term broadens the parameter ranges for which weak solutions exist compared to prior results, and under additional conditions the local minimum coincides with a least-energy solution. Overall, the paper demonstrates that nonlocal perturbations can positively affect the existence theory for critical 4D Kirchhoff problems with logarithmic perturbations, enriching the variational landscape for such nonlocal, sign-changing nonlinearities.

Abstract

In this paper, a critical Kirchhoff equation with a logarithmic type subcritical term is considered in a bounded domain in $\mathbb{R}^4$. We view this problem as a critical elliptic equation with a nonlocal perturbation, and investigate how the nonlocal term affects the existence of weak solutions to the problem. By means of Ekeland's variational principle, Brézis-Lieb's lemma and some convergence tricks for nonlocal problems, we show that this problem admits a local minimum solution and a least energy solution under some appropriate assumptions on the parameters. Moreover, under some further assumptions, the local minimum solution is also a least energy solution. Compared with the ones obtained in [3] and [8], our results show that the introduction of the nonlocal term enlarges the ranges of the parameters such that the problem admits weak solutions, which implies that the nonlocal term has a positive effect on the existence of weak solutions.

Figures (3)

• Figure : $(1) \ b\geq\frac{1}{S^2}$
• Figure : $(1) \ b\geq\frac{1}{S^2}$
• Figure : $(2) \ 0<b<\frac{1}{S^2}$ or $\frac{1}{2S^2}\leq b<\frac{1}{S^2}$

Theorems & Definitions (3)

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