Radial symmetric stationary solutions for a MEMS type reaction–diffusion equation with spatially dependent nonlinearity

We consider the radial symmetric stationary solutions of u_t= Δu- | x| ^qu^-p. We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for p∈ 2 N, q∈ R. Moreover, we give the asymptotic behavior of u(r) and u^′(r) at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at r= 0 for p∈ N and q≥ - 2. In fact, the behavior of solutions for q> - 2 and q= - 2 are different. In each case of q= - 2 and q> - 2 , we derive the asymptotic behavior for r→ 0 and r→ ∞. These facts are studied by applying the Poincaré compactification and basic theory of dynamical systems.