Abstract
A sufficient and necessary condition for existence of solution for the boundary blow-up problem in one dimensional case is obtained. This problem can be seen as the Keller-Osserman conjecture, which comes from the study on elliptic equations.
Highlights
The boundary blow-up problem u(x) = f (u(x)), x ∈, ( . )u|∂ = +∞, where is a bounded domain in RN (N ≥ ), arises in many fields, such as the theory of automorphic functions and Riemann surfaces of constant negative curvature, the study of the electric potential in a glowing hollow metal body, etc
In, this type of problem was first studied by Bieberbach
The condition ( . ) plays an important role in the study of the boundary blow-up problem, it was first proposed by Keller [ ] and Osserman [ ], lately, this condition was called the Keller-Osserman condition
Summary
Keller-Osserman conjecture When N = , the conclusion of Theorem A is true. They gave some necessary, sufficient conditions for the existence of nonnegative solutions. As they said, ‘The gap between the class of functions that satisfy the necessary condition but not the sufficient condition is quite small.’. Wang [ ] used the same method as Anuradha’s and generalized the results of Anuradha’s; he obtained a more suitable condition for the existence of solutions for ). The gap between the class of functions that satisfy the necessary condition but not the sufficient condition becomes smaller than that of Anuradha’s, but we still have a distance to Keller-Osserman’s conjecture. ) has a positive solution if and only if the Keller-Osserman condition Lemma If u(t) is a solution of ( . ), there exists only one point t ∈ ( , ) such that u (t )
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