Existence and nonexistence of solutions to a critical biharmonic equation with logarithmic perturbation

TL;DR

This paper investigates the existence and nonexistence of solutions to a critical biharmonic equation with a logarithmic perturbation, revealing new phenomena due to the perturbation's sign uncertainty using variational methods.

Contribution

It establishes conditions for existence and nonexistence of solutions to a biharmonic problem with a logarithmic term, highlighting novel phenomena caused by the perturbation.

Findings

Existence of at least one nontrivial weak solution under certain parameters.

Nonexistence results under specific conditions.

New phenomena due to the logarithmic perturbation's sign uncertainty.

Abstract

In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases} \end{eqnarray*} is considered, where $\Omega\subset \mathbb{R}^{N}$ is a bounded smooth domain with $N\geq5$. Some interesting phenomenon occurs due to the uncertainty of the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at lest one nontrivial weak solution under some appropriate assumptions of $\lambda$ and $\mu$. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced.