Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems

Abstract

We show that entire large positive radial solutions exist for the semilinear elliptic system 2u= p(|x|) v : , 2v=q(|x|) u ; on R N , N 3, for positive : and ;, provided that the nonnegative functions p and q are continuous, c-positive, and satisfy the decay conditions 0 tp(t) dt< and 0 tq(t) dt< for : and ; greater than unity, and 0 tp(t) dt= and 0 tq(t) dt= if neither : nor ; is greater than one.

Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

“…Suppose 1 > α ≥ β > 0 and that p and q have a fast decay in the sense of (5). Then every bounded entire solution u v of system (2) is bounded in R N .…”

“…See, for example, [1,[3][4][5][8][9][10][11][12] and the references therein. Most of these results have to do with the nonexistence of positive solutions, the existence of radial solutions, or the asymptotic behavior of the solutions.…”

“…In particular, in [1], the nonexistence of positive radial solutions was shown for the system where f u v ≥ au k v p and g u v ≥ bu q v t with k t > 1, p q ≥ 0, and a b both positive constants. In [5], the authors consider the system…”

“…We are unable to prove the existence of large entire solutions as suggested by the results in [5], whose proofs rely on the radial symmetry assumption in the system. For the single equation…”

Positive solutions for some competitive elliptic systems

“…For some recent results on the qualitative analysis and the applications of positive solutions of nonlinear elliptic systems in both bounded and unbounded domains we refer to [10,11,[13][14][15][16][17][18][19][20]26] and the references therein.…”