Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

Abstract

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $Δ^2 u = λf(u)$ on a general bounded domain $Ω$ in $ \IR^N$, with the Navier boundary condition $ u=Δu =0 $ on $ \pOm$. Here $ λ$ is a positive parameter and $f$ is a non-decreasing nonlinearity with $f(0)=1$. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity $f$, the extremal solution $ u^*$ is smooth provided $N\leq 5$.