Critical exponent of a simple model of spot replication

This paper is concerned with a semilinear elliptic inhomogeneous equationΔu−u+(1+a|x|q)up=0 introduced in Chen and Kolokolnikov (2012) [2] as a simple prototype of self-replication in more complex reaction-diffusion systems. Under certain conditions on p, q, it was previously shown by Chen-Kolokolnikov that the equation has no radial ground state solution when the control parameter a is increased above some threshold. This property is important for the existence of a saddle-node bifurcation proposed in the Nishiura-Ueyema conditions, which is believed to be necessary for an initiation of a self-replication event. In this paper, we generalize Chen-Kolokolnikov's result to non-radial positive solutions by proving a Liouville-type nonexistence theorem. Furthermore we derive a local version of this nonexistence theorem for solutions defined on a bounded ball. Our result indicates that critical values of q derived in Ding and Ni (1986) [3] are also crucial for the existence and nonexistence problem of positive solutions when the space dimension N≥3.