Liouville type result and long time behavior for Fisher-KPP equation with sign-changing and decaying potentials

This paper concerns the Liouville type result for the general semilinear elliptic equation a_ij(x)∂_iju(x)+Kq_i(x)∂_iu(x)+f(x,u(x))=0a.e. in R^N, where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow f_s(x,0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (a_ij) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K>K^⋆, where K^⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14], to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not.