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

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 $f:[0,a_f)\to {\mathbb{R}}_{+}$ $(0<a_f⩽\infty )$ is a smooth, increasing, convex nonlinearity such that $f(0)>0$ and which blows up at $a_f$. Let $$0<{\tau }_{-}:=\underset{t\to a_f}{\mathrm{liminf}}\frac{f(t)f"(t)}{f^{\prime }(t{)}^2}\le {\tau }_{+}:=\underset{t\to a_f}{\mathrm{limsup}}\frac{f(t)f"(t)}{f^{\prime }(t{)}^2}<2.$$ We show that if $u_m$ is a sequence of semistable solutions correspond to ${\lambda }_m$ satisfy the stability inequality $$\sqrt{{\lambda }_m}{\int }_{\Omega }\sqrt{f^{\prime }(u_m)}{\varphi }^{2}dx\le {\int }_{\Omega }|\nabla \varphi {|}^{2}dx,\text{for all}\varphi \in H_0^1(\Omega ),$$ then ${\sup }_m||u_m|{|}_{L^{\infty }(\Omega )}<a_f$ for $n<\frac{4{\alpha }_{\ast }(2-{\tau }_{+})+2{\tau }_{+}}{{\tau }_{+}}\max \{1,{\tau }_{+}\},$ where ${\alpha }^{\ast }$ 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 $n\le 12$ when $\tau \le 1$, and for $n\le 7$ when $\tau \le 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{\ast }(x)={\lim }_{\lambda \uparrow {\lambda }^{\ast }}{u}_{\lambda }(x),$ where ${\lambda }^{\ast }$ is the extremal parameter of the eigenvalue problem.

14 pages, submitted. In this version just a typo is removed from the title

References in corpus (2)

• The Gel'fand problem for the biharmonic operator
• A priori estimates for semistable solutions of semilinear elliptic equations