Note on singular semilinear elliptic equations (Q1210057)

This note deals with the existence of positive entire solution of the following singular semilinear elliptic equation \[ -\Delta u+c(x)u= p(x)u^{-\gamma}, \quad \text{in } \mathbb{R}^ n, \quad n\geq 3,\quad \gamma>0,\tag{1} \] where \(c\), \(p\) are locally Hölder continuous in \(\mathbb{R}^ n\) with exponent \(0<\theta<1\) and \(c(x)>0\) in \(\mathbb{R}^ n\). An entire solution of the problem is a function \(u\) in \(C^{2+\theta} (\mathbb{R}^ n)\) satisfying the equation in (1) pointwise in \(\mathbb{R}^ n\). The author proves the existence of positive entire solution with uniform positive limits at infinity. Moreover, under a suitable condition on the function \(p\), there exists an entire solution decaying to 0 at infinity.