Explosive solutions of a class of semilinear elliptic equation (Q2774276)

The authors studied the semilinear elliptic equation \(\Delta u= p(x)f(u)\), where \(f(s)\) is a nonnegative continuous differentiable function in \((0,\infty)\) which is monotone increasing, and \(\lim_{s\to\infty} f(s)= 0\), \(\lim_{s\to\infty} {f(s)\over s}= k\) \((k<\infty)\), \(p(x)\) is a nonnegative locally Hölder continuous function in \(\mathbb{R}^N\) \((N\geq 3)\). Using an iteration method, the authors proved that when \(p(x)= p(|x|)\), the equation has a entire explosive solution if and only if \(\int^\infty_0 tp(t) dt= \infty\); when \(p(x)\) satisfies \(\int^\infty_0 t\varphi(t) dt<\infty\), the equation has a entire bounded solution, where \(\varphi(t)= \max_{|x|=t} p(x)\).