Existence of solution for Kirchhoff model problems with singular nonlinearity

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We study the fourth order Kirchhoff equation Δ2u−(a+b∫Ω|∇u|2)γΔu=f(u) in Ω with −Δu>0 and u>0 in Ω, and Δu=u=0 on ∂Ω, where f(t)=α1tθ+λtq+μt+g(t) for t≥0, g has subcritical growth, α>0, λ>0, μ≥0, 0<θ<1, 0<q<1, γ≥0, a>0, b≥0. We use the Galerkin projection method to show the existence of solution under some boundedness restriction on α,λ,μ. In some cases we study the behavior of the norm of the solution u as λ→0 and as λ→∞. Similar issues are addressed for the equation (a+b∫Ω|∇u|2)γΔ2u−ϱΔu=f(u), ϱ≥0.