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

Abstract

In this paper, the following critical biharmonic elliptic problem{Δ2u=λu+μuln⁡u2+|u|2⁎⁎−2u,x∈Ω,u=∂u∂ν=0,x∈∂Ω is considered, where Ω⊂RN is a bounded smooth domain with N≥5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. 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.