Non-existence of radially symmetric non-negative solutions for a class of semi-positone problems (Q1174381)

The authors consider the Dirichlet problem \[ -\Delta u(x)=\lambda f(u(x)) \hbox { in } B_ 1(0), \quad u(x)=0 \hbox{ on }\partial B_ 1(0) \leqno (*) \] under the conditions \(\lambda>0\), \(f'\geq0\), \(f(0)<0\), \(\liminf_{u\to\infty}(f(u)/u^ \alpha)>0\) for some \(\alpha>1\). They show that for fixed nonlinearity \(f\), there exists a number \(\lambda_ 0>0\) such that for every \(\lambda\geq\lambda_ 0\), (*) has no nonnegative radially symmetric solution. That means that e.g. in \textit{P. Quittner} [Comment. Math. Univ. Carol. 30, No. 3, 579-585 (1989; Zbl 0698.35057)] the smallness assumption on \(\lambda f(0)\) is necessary to prove existence.