Positive and oscillatory radial solutions of semilinear elliptic equations

doi:10.1155/S1048953397000105

Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 1, Pages 95-108
http://dx.doi.org/10.1155/S1048953397000105

Positive and oscillatory radial solutions of semilinear elliptic equations

Shaohua Chen, ¹ William R. Derrick, ¹ and Joseph A. Cima ²

¹University of Montana, Department of Mathematics, Missoula 59812, MT, USA
²University of North Carolina, Department of Mathematics, Chapel Hill 27599, NC, USA

Received 1 September 1995; Revised 1 February 1996

Copyright © 1997 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that the nonlinear partial differential equation Δ u + f ( u ) + g ( | x | , u ) = 0 , in ℝ n , n ≥ 3 , with u ( 0 ) > 0 , where f and g are continuous, f ( u ) > 0 and g ( | x | , u ) > 0 for u > 0 , and lim u → 0 + f ( u ) u q = B > 0 , for 1 < q < n / ( n − 2 ) , has no positive or eventually positive radial solutions. For g ( | x | , u ) ≡ 0 , when n / ( n − 2 ) ≤ q < ( n + 2 ) / ( n − 2 ) the same conclusion holds provided 2 F ( u ) ≥ ( 1 − 2 / n ) u f ( u ) , where F ( u ) = ∫ 0 u f ( s ) d s . We also discuss the behavior of the radial solutions for f ( u ) = u 3 + u 5 and f ( u ) = u 4 + u 5 in ℝ 3 when g ( | x | , u ) ≡ 0 .