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

TL;DR

This paper investigates the regularity of extremal solutions to semilinear elliptic equations with non-convex nonlinearities on general domains, extending known results to higher dimensions based on the nonlinearity's growth conditions.

Contribution

It establishes new regularity results for extremal solutions in higher dimensions depending on the asymptotic behavior of the nonlinearity, beyond the previously known two-dimensional case.

Findings

Extremal solutions are bounded in dimensions up to 6 under certain conditions.

Boundedness of solutions in dimensions up to 9 for specific nonlinearity growth rates.

Solutions belong to Sobolev space H^1_0 for dimensions ≥ 2 if the nonlinearity's derivative satisfies certain limits.

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 $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $\Omega$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabr\'{e} \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<\beta_{-}:=\liminf_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}<\infty$$ where $F(t)=\int_{0}^{t}f(s)ds$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 6$. Also, if $\beta_{-}=\beta_{+}>\frac{1}{2}$ or $\frac{1}{2}<\beta_{-}\leq\beta_{+}<\frac{7}{10}$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 9$. Moreover, if $\beta_{-}>\frac{1}{2}$ then $u^{*}\in H^{1}_{0}(\Omega)$ for $n\geq 2$.