Positive and oscillatory radial solutions of semilinear elliptic equations. (1997) - Chen, Shaohua; Derrick, William R.; Cima, Joseph A.
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 .