Dominated Weak Solutions for a Second Order Ordinary Differential Equation with Boundary Conditions in (0,∞)

In this paper, we study the existence of dominated weak solutions u of the boundary-value problem ( - g ( t ) ( u ( t ) ′ ) γ ) ′ = f ( t , u ( t ) ) , in ( 0 , ∞ ) , u ( 0 ) = 0 , u ( ∞ ) = 0 , where f is a Carathéodory function, g is a positive function, and γ is an odd positive integer. More prec...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2022-06, Vol.19 (3), Article 128
Main Author: Villa-Morales, José
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary value problems
Differential equations
Mathematics
Mathematics and Statistics
Ordinary differential equations
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Summary:In this paper, we study the existence of dominated weak solutions u of the boundary-value problem ( - g ( t ) ( u ( t ) ′ ) γ ) ′ = f ( t , u ( t ) ) , in ( 0 , ∞ ) , u ( 0 ) = 0 , u ( ∞ ) = 0 , where f is a Carathéodory function, g is a positive function, and γ is an odd positive integer. More precisely, given barrier functions h 0 and h 1 in L p ( 0 , ∞ ) , 1 ≤ p < ∞ , we prove the existence of weak solutions in a Sobolev-type space such | u | ≤ h 0 a.e. and | u ′ | ≤ h 1 a.e. The Palais-Smale condition is not assumed and no reflexivity property is applied, instead a sort of sequential compactness in L p ( 0 , ∞ ) is used to show the weak existence of solutions.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-022-02031-4