The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators

被引:16
作者
Marcelli, Cristina [1 ]
机构
[1] Univ Politecn Marche, Dipartimento Ingn Ind & Sci Matemat, I-60131 Ancona, Italy
来源
BOUNDARY VALUE PROBLEMS | 2013年
关键词
boundary value problems; unbounded domains; heteroclinic solutions; nonlinear differential operators; p-Laplacian operator; Phi-Laplacian operator; PHI-LAPLACIAN; PROBLEM; (PHI(U'))'=F(T; U; U'); HETEROCLINIC SOLUTIONS; PERIODIC PROBLEMS; EXISTENCE;
D O I
10.1186/1687-2770-2013-252
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The paper deals with the existence and non-existence of solutions of the following strongly nonlinear non-autonomous boundary value problem: (P) {(a(t, x(t)) Phi (x'(t)))' = f (t, x(t), x' (t)) a. e. t is an element of R, x(-infinity) = nu(-), x(+infinity) = nu(+) with nu(-) < nu(+), where Phi : R -> R is a general increasing homeomorphism, with Phi (0) = 0, a is a positive, continuous function and f is a Caratheodory nonlinear function. The same problem was already studied in the case when |f (t, x, y)/Phi (y)| -> 0 as y -> 0 in the recent paper (Marcelli in Electron. J. Differ. Equ. 2012: 171, 2012), where sharp sufficient conditions for the existence or non-existence of solutions were established. In particular, it was proved that neither the behavior of the functions a(t,center dot) and f (t,center dot,y) nor the boundary data nu(-), nu(+) influence the solvability of problem (P). We herein study the critical case when |f (t, x, y)| similar to |Phi(y)| as |y| -> 0, focusing on the role played by the dependence on x of the functions a and f and by the boundary data nu(-), nu(+) by means of an explicit link between them and the other parameters of the differential equation.
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页数:14
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