Abstract
In this paper, we give the exact asymptotic behavior of the unique positive solution to the following singular boundary value problem 8 1 (Au 0 ) 0 = p(x)g(u), x2 (0, 1),
Highlights
In this paper, we give the exact asymptotic behavior near the boundary of the unique positive solution to the following singular problem= p(x)g(u), x ∈ (0, 1), (1.1)u > 0, in (0, 1), subject to the boundary conditions lim (Au )(x) = 0, u(1) = 0. (1.2) x→0+The functions A, p and g satisfy the following assumptions. (H1)A is a continuous function on [0, 1), positive and differentiable on (0, 1) such that and limx→1 (x−1)A (x) A(x)α (H2) p is a nonnegative continuous function in (0, 1) satisfying p(x) lim inf x→1 h(1
Under some appropriate conditions on the function A, they proved the existence of a unique positive solution u in C([0, 1]) ∩ C2((0, 1)) to (1.3) and gave estimates on such a solution
Let L be a function in K defined on
Summary
We give the exact asymptotic behavior near the boundary of the unique positive solution to the following singular problem. In [15, Theorem 5], Mâagli and Masmoudi investigated equation (1.3) with boundary value conditions u (0) = u(1) = 0 They supposed that f is a nonnegative continuous function on (0, 1) × (0, ∞) and nonincreasing with respect to the second variable. Under some appropriate conditions on the function A, they proved the existence of a unique positive solution u in C([0, 1]) ∩ C2((0, 1)) to (1.3) and gave estimates on such a solution In particular they extended some results of [1, 2] and [19]. Our aim in this paper is to establish the exact boundary behavior of the unique solution to problem (1.1)–(1.2).
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