Regularity and uniqueness results for second-order nonlinear eigenvalue problems

We consider the following nonlinear eigenvalue problems {−Δu=λf(x)(1−u)px∈Ω;u=0x∈∂Ω, where p>0,λ≥0,Ω⊂Rnp>0,λ≥0,Ω⊂Rn is smooth bounded domain and f∈C(Ω̄) is a nonnegative function. Our study is motivated by a simple Micro-Electromechanical Systems (MEMS) device model. We find some new phenomena about the regularity results by the classification of entire solutions. Besides, we also consider the uniqueness of solutions by Rellich–Pohozaev type estimates.