Maximum principle methods for semilinear elliptic problems

Equations of the form $\Delta u+f(u)=0$ have been extensively studied in a bounded domain $\Omega$ of $R\sp N$, especially in finding steady states of reaction diffusion systems. In this paper, we extend the symmetry result given by Gidas, Ni, and Nirenberg to certain types of systems.

We also study the singular equation $\Delta u+p(x)u\sp{-\gamma}=0$ in $R\sp{n}$. We show that the above equation has a bounded, positive, $C\sp{2+\alpha}$ entire solution $u(x)$ vanishing at $\infty$ at the rate of at least $r\sp{q(2-n)}$, $0 < q < 1$, if $p(x)$ satisfies the following conditions: (1) $p(x) \in C\sbsp{loc}{\alpha} (R\sp n), n \geq 3, p(x) > 0, x \in R\sp n\\\{0\}$; (2) there exists $C > 0$, such that $C\phi(\vert x\vert) \leq \phi(\vert x\vert), \phi(x) = max\sb{\vert x\vert=t}p(x)$, $0 \leq t \leq \infty$; (3) $\int\sbsp{1}{\infty} t\sp{n-1+\gamma(n-2)}\phi(t)dt < \infty$.