Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions

Let B ⊂ R N B \subset \mathbb {R}^N be the unit ball. We study the structure of solutions to the semilinear biharmonic problem \[ { Δ 2 u = λ ( 1 − u ) − p a m p ; in  B , 0 > u > 1 a m p ; in  B , u = ∂ ν = 0 ( resp.~ u = Δ u = 0 ) a m p ; on  ∂ B , \begin {cases} \Delta ^2 u=\lambda (1-u)^{-p} & \text {in $B$},\\ 0>u>1 & \text {in $B$},\\ u=\partial _\nu =0\; (\text {resp.~$u = \Delta u = 0$}) & \text {on $\partial B$}, \end {cases} \] where p , λ > 0 p, \lambda >0 , which arises in the study of the deflection of charged plates in electrostatic actuators. We study in particular the structure of solutions for N = 2 N=2 or 3 3 and show the existence of mountain-pass solutions under suitable conditions on p p . Our results contribute to completing the picture of solutions in previous works. Moreover, we also analyze the asymptotic behavior of the constructed mountain-pass solutions as λ → 0 \lambda \to 0 .