Multiple positive solutions of a singular Emden-Fowler type problem for second-order impulsive differential systems
The authors consider the boundary value problem with impulses \[ \begin{aligned}&u''(t) + \lambda h_1(t)f(u(t),v(t)) = 0, \quad t \in (0,1), \;t\neq t_1,\\ &v''(t) + \mu h_2(t)g(u(v),v(t)) = 0, \quad t \in (0,1), \;t\neq t_1,\\ &\triangle u_{|t=t_1} = I_u(u(t_1)), \quad \triangle v_{|t=t_1} = I_v(v(t_1)),\\ &\triangle u'_{|t=t_1} = N_u(u(t_1)), \quad \triangle v'_{|t=t_1} = N_v(v(t_1)),\\ &u(0) = a \geq 0, \quad v(0) = b \geq 0, \quad u(1) = c \geq 0, \quad v(1) = d \geq 0 \end{aligned} \] where \(\lambda\), \(\mu>0\), \(f\), \(g\), \(I_u\), \(I_v\), \(N_u\), \(N_v\), \(h_i\) are continuous, \(h_i\) may have a singularity at \(t=0\) and/or \(t=1\). Sufficient conditions ensuring the existence of at least one positive solution as well as two positive solutions. The results are obtained via the lower and upper solutions technique and fixed point index theorems.
scientific article; zbMATH DE number 5825758
scientific article