Lp-Solvability of a fourth-order boundary-value problem at resonance

摘要

Let Ω be a bounded domain in Rn with smooth boundary Γ. We obtain existence results for the solutions of the biharmonic boundary value problems, -Δ2u + ⋋21u + g(x, u) = f, in Ω, u = Δu = 0 on Γ; -Δ2u + g(x, u) = f, Ω, ∂u/∂n = ∂(Δu)/∂n = 0, on Γ when g(x, u) has linear growth in u and f is in certain subclass of Lp(Ω). In the first problem, ⋋1 is the first eigenvalue of the eigenvalue problem -Δu = ⋋u in Ω and u = 0 on Γ.