Steady state solutions for the Gierer-Meinhardt system in the whole space

Abstract

We are concerned with the study of positive solutions to the Gierer-Meinhardt system{−Δu+λu=upvq+ρ(x)in RN,N≥3,−Δv+μv=umvsin RN, which satisfy u(x),v(x)→0 as |x|→∞. In the above system p,q,m,s>0, λ,μ≥0 and ρ∈C(RN), ρ≥0. It is a known fact that posed in a smooth and bounded domain of RN, the above system subject to homogeneous Neumann boundary conditions has positive solutions if p>1 and σ=mq(p−1)(s+1)>1. In the present work we emphasize a different phenomenon: we see that for λ,μ>0 large, positive solutions with exponential decay exist if 0<σ≤1. Further, for λ=μ=0 we derive various existence and nonexistence results and underline the role of the critical exponents p=NN−2 and p=N+2N−2.