Invariant criteria for existence of bounded positive solutions

Feng Zhou, Dong Ye
2004 Discrete and Continuous Dynamical Systems. Series A  
We consider semilinear elliptic equations ∆u ± ρ(x)f (u) = 0, or more generally ∆u + ϕ(x, u) = 0, posed in R N (N ≥ 3). We prove that the existence of bounded positive entire solutions is closely related to the existence of bounded solution for ∆u + ρ(x) = 0 in R N . Many sufficient conditions which are invariant under the isometry group of R N are established. Our proofs use the standard barrier method, but our results extend many earlier works in this direction. Our ideas can also be applied
more » ... an also be applied for the existence of large solutions and for the system cases.
doi:10.3934/dcds.2005.12.413 fatcat:5viamud5qzhkri5bmnodw7pcou