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 ${\Delta }^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $u=\Delta u=0$ on $ \pOm$. Here $\lambda$ 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 $u^{\ast }$ is smooth provided $N\le 5$.

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