Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

Abstract

We consider the semilinear elliptic equation $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^n$ with Dirichielt boundary condition, where $f$ is a $C^1$ positive and nondeccreasing function in $[0,\infty )$ such that $\frac{f(t)}{t}\to \infty$ as $t\to \infty$. When $\Omega$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{\ast }$ is known only for $n=2$, established by X. Cabré \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<{\beta }_{-}:=\underset{t\to \infty }{\mathrm{liminf}}\frac{f^{\prime }(t)F(t)}{f(t{)}^2}\le {\beta }_{+}:=\underset{t\to \infty }{\mathrm{limsup}}\frac{f^{\prime }(t)F(t)}{f(t{)}^2}<\infty$$ where $F(t)={\int }_0^{t}f(s)ds$, then $u^{\ast }\in L^{\infty }(\Omega )$, for $n\le 6$. Also, if ${\beta }_{-}={\beta }_{+}>\frac{1}{2}$ or $\frac{1}{2}<{\beta }_{-}\le {\beta }_{+}<\frac{7}{10}$, then $u^{\ast }\in L^{\infty }(\Omega )$, for $n\le 9$. Moreover, if ${\beta }_{-}>\frac{1}{2}$ then $u^{\ast }\in H_0^1(\Omega )$ for $n\ge 2$.