On an overdetermined eigenvalue problem with mems operator

Abstract

We consider an overdetermined eigenvalue problem related to the MEMS operator given by Lτ := ∆2 − τ∆ on a smooth bounded domain Ω ⊂ R N , N ≥ 2. We give radial solutions on balls. Moreover, we establish a symmetry result with respect to operator Lτ , that is, under some hypotheses, we show that if a solution does exist to the overdetermined eigenvalue problem, then the domain Ω must be a ball.