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

TL;DR

This paper investigates the regularity of extremal solutions to a fourth order nonlinear eigenvalue problem on general domains, establishing smoothness results for dimensions up to five under certain conditions.

Contribution

It provides new pointwise bounds and energy estimates demonstrating the smoothness of extremal solutions for convex, superlinear nonlinearities in dimensions up to five.

Findings

Extremal solutions are smooth for N ≤ 5.

General pointwise bounds are established.

Energy estimates support regularity results.

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^*$ is smooth provided $N\leq 5$.