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

Abstract

We consider the fourth order problem $Δ^{2}u=λf(u)$ on a general bounded domain $Ω$ in $R^{n}$ with the Navier boundary condition $u=Δu=0$ on $\partial Ω$. Here, $λ$ 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 $$0<τ_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $λ_{m}$ satisfy the stability inequality $$ \sqrt{λ_{m}}\int_Ω\sqrt{f'(u_{m})}ϕ^{2}dx\leq \int_Ω|\nablaϕ|^{2}dx, ~~\text{for all}~ϕ\in H^{1}_{0}(Ω),$$ then $\sup_{m} ||u_{m}||_{L^{\infty}(Ω)}<a_{f}$ for $n< \frac{4α_{*}(2-τ_{+})+2τ_{+}}{τ_{+}}\max \{1, τ_{+}\},$ where $α^{*}$ is the largest root of the equation $$(2-τ_{-})^{2} α^{4}- 8(2-τ_{+})α^{2}+4(4-3τ_{+})α-4(1-τ_{+})=0.$$ In particular, if $τ_{-}=τ_{+}:=τ$, then $\sup_{m} ||u_{m}||_{L^{\infty}(Ω)}<a_{f}$ for $n\leq12$ when $τ\leq 1$, and for $n\leq7$ when $τ\leq 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x)=\lim_{λ\uparrowλ^{*}}u_λ(x),$ where $λ^*$ is the extremal parameter of the eigenvalue problem.