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

We consider the fourth order problem $Δ^{2}u = λ f(u)$ on a general bounded domain $Ω$ in $R^{n}$ with the Navier boundary condition $u = Δ u = 0$ on $\partial Ω$. Here, $λ$ is a positive parameter and $ f:[0, a_{f}) \to \Bbb{R}_{+} $ $ \left( {0 < {a_f} \le \infty } \right)$ is a smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ {a_f} $. Let

$0<τ_{-}: = \liminf\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}≤q τ_{+}: = \limsup\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}<2.$

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})}\phi ^{2}dx≤\int{{_{Ω}}}|\nablaφ|^{2}dx, ~~\text{for all}~\phi ∈ H^{1}_{0}(Ω), $

then $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ 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 $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ for $n≤12$ when $τ≤ 1$, and for $n≤7$ when $τ≤ 1.57863$. 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.