Singular positive solutions for a fourth order elliptic problem in \(\mathbb R^N\) (Q651963)

The authors study semilinear elliptic problems involving the bi-Laplacian operator, of the type \[ \Delta^2 u - c_1\Delta u + c_2 u = u^p + f, \] where \(N\geq 5\), \(\kappa > 0\), \(m \in \mathbb{N}\), \(\alpha_i>0\), \(c_1, c_2 > 0\) and \(f = \kappa \sum_{i=1}^{m} \alpha_i \delta_{a_i}\) with \(\delta_{a_i}\) denoting the delta distribution supported at \(a_i\). Throughout, \(p \in (1, N/(N-4))\) and \(c_1^2 - 4c_2 \geq 0\), so that for \(0 < t_1 \leq t_2\) solutions of \(t^2 -c_1t + c_2 = 0\), the equation can be written as the system \[ -\Delta u + t_1 u = v, \;\;-\Delta v + t_2 v = u^p + \kappa f. \] It is shown that there exists \(\kappa^* > 0\) so that for \(\kappa > \kappa^*\) the problem admits no positive solution, while for \(\kappa \in (0, \kappa^*)\) there exist two positive solutions \(u_\kappa < u^\kappa\) on \(\mathbb{R}^N \setminus \bigcup_{i=1}^m \{a_i\}\). Asymptotic estimates are also given for \(u_\kappa\) and \(u^\kappa\), as \(x \to a_i\) and as \(|x| \to \infty\).