In this paper we study in detail questions concerning boundedness and asymptotic behavior for extremals of the variational problem δ∫Jg(r)[G(u′)−F(r,u)]dr=0, where J is an open interval; g,G, and F are C1 functions; and g(r)>0, F(r,0)=0, G(p)>0, G(0)=0, G is strictly convex. We first give conditions that ensure that extremals can be continued across the entire interval. Next, boundedness and asymptotic behavior of extremals is discussed. For the semi-infinite interval, we present hypotheses that imply convergence of the extremal and its derivative to zero as r→∞. Part II of the paper contains an analysis of the special case of the degenerate Laplace operator, where G(p)=|p|m/m. In some cases, the extremal decays algebraically to zero, while in others there is oscillatory behavior. Much use is made throughout of a family of Lyapunov functions that is obtained from a fundamental identity for these extremals. The results in this paper augment and generalize earlier work of Z. Artstein and E. F. Infante [Quart. Appl. Math. 34 (1976/77), 195-199], and J. J. Levin and J. A. Nohel [Arch. Rational Mech. Anal. 5 (1960), 194-211].
Continuation and limit properties for solutions of strongly nonlinear second order differential equations
PUCCI, Patrizia;
1991
Abstract
In this paper we study in detail questions concerning boundedness and asymptotic behavior for extremals of the variational problem δ∫Jg(r)[G(u′)−F(r,u)]dr=0, where J is an open interval; g,G, and F are C1 functions; and g(r)>0, F(r,0)=0, G(p)>0, G(0)=0, G is strictly convex. We first give conditions that ensure that extremals can be continued across the entire interval. Next, boundedness and asymptotic behavior of extremals is discussed. For the semi-infinite interval, we present hypotheses that imply convergence of the extremal and its derivative to zero as r→∞. Part II of the paper contains an analysis of the special case of the degenerate Laplace operator, where G(p)=|p|m/m. In some cases, the extremal decays algebraically to zero, while in others there is oscillatory behavior. Much use is made throughout of a family of Lyapunov functions that is obtained from a fundamental identity for these extremals. The results in this paper augment and generalize earlier work of Z. Artstein and E. F. Infante [Quart. Appl. Math. 34 (1976/77), 195-199], and J. J. Levin and J. A. Nohel [Arch. Rational Mech. Anal. 5 (1960), 194-211].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
