Multiplicity of positive radial solutions for an elliptic system on an annulus
The authors study the existence of positive radial solutions of the elliptic system \[ \Delta u + \lambda k_1(|x |) f(u,v)=0, \quad \Delta v + \lambda k_2(|x |) g(u,v)=0 \text{ in } \Omega, \] \[ \alpha_1 u + \beta_1 \frac{\partial u}{\partial n}=0, \quad \alpha_2 v + \beta_2 \frac{\partial v}{\partial n}=0 \text{ on } |x |=R_1, \] \[ \gamma_1 u + \delta_1 \frac{\partial u}{\partial n}=0, \quad \gamma_2 v + \delta_2 \frac{\partial v}{\partial n}=0 \text{ on } |x |=R_2, \] where \(\alpha_i\), \(\beta_i\), \(\gamma_i\), \(\delta_i \geq 0\) and \(\gamma_i \beta_i + \alpha_i \gamma_i + \alpha_i \delta_i >0\) for \(i=1, 2.\) Under suitable conditions on the functions that define the system, they prove that there exist \(\lambda^* >0\) for which this problem has at least two positive solutions for \(0< \lambda < \lambda^*\), at least one positive solution for \(\lambda=\lambda^*\) and no positive solution for \(\lambda > \lambda^*\). In the proof degree theory, Green's functions properties and existence results via lower and upper solutions are used.
scientific article; zbMATH DE number 1513296