Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

TL;DR

This paper investigates the regularity of extremal solutions to a fourth-order elliptic problem with general nonlinearities, establishing bounds on solutions based on domain dimension and nonlinearity behavior, thus extending understanding of solution regularity.

Contribution

It provides new regularity results for extremal solutions of fourth-order elliptic problems with general nonlinearities, including explicit bounds depending on nonlinear growth and domain dimension.

Findings

Solutions are bounded in L-infinity norm under certain dimension and nonlinearity conditions.

Extremal solutions are regular (bounded) in dimensions up to 12 for specific nonlinearity parameters.

Derived explicit bounds involving roots of a polynomial related to the nonlinearity's growth.

Abstract

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. Let $$0<\tau_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq \tau_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $\lambda_{m}$ satisfy the stability inequality $$ \sqrt{\lambda_{m}}\int_{\Omega}\sqrt{f'(u_{m})}\phi^{2}dx\leq \int_{\Omega}|\nabla\phi|^{2}dx, ~~\text{for all}~\phi\in H^{1}_{0}(\Omega),$$ then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n< \frac{4\alpha_{*}(2-\tau_{+})+2\tau_{+}}{\tau_{+}}\max \{1, \tau_{+}\},$ where $\alpha^{*}$ is the largest root of the equation $$(2-\tau_{-})^{2} \alpha^{4}- 8(2-\tau_{+})\alpha^{2}+4(4-3\tau_{+})\alpha-4(1-\tau_{+})=0.$$ In particular, if $\tau_{-}=\tau_{+}:=\tau$, then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n\leq12$ when $\tau\leq 1$, and for $n\leq7$ when $\tau\leq 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x)=\lim_{\lambda\uparrow\lambda^{*}}u_{\lambda}(x),$ where $\lambda^*$ is the extremal parameter of the eigenvalue problem.