multiplicity of positive radially symmetric solutions for a quasilinear biharmonic equation in the plane

this paper is concerned with the multiplicity of positive radially symmetric solutions of the dirichlet boundary value problem for the following two-dimensional quasilinear biharmonic equation

δ(|δu|^p−2δu) = λg(x)f (u), x ∈ b₁, where b₁ is the unit ball in the plane. we apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions. we have found that there exists a threshold λ^∗ < +∞, such that if λ > λ^∗, then the problem has no positive radially symmetric solution; while if 0 < λ ≤ λ^∗, then the problem admits at least one positive radially symmetric solution. especially, there exist at least two positive radially symmetric solutions for 0 < λ < λ^∗.