Abstract
A class of elliptic boundary value problem in an exterior domain is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where the variables of nonlinear term f(s, u) need not to be separated. Several new theorems on the existence and multiplicity of positive radial solutions are obtained by means of fixed point index theory. Our conclusions are essential improvements of the results in Lan and Webb (1998), Lee (1997), Mao and Xue (2002), Stańczy (2000), and Han and Wang (2006).
Highlights
The existence and multiplicity of positive radial solution for the following elliptic boundary value problem−Δu = f (|x|, u) for |x| > 1, x ∈ Rn, n ⩾ 3, u = 0 for |x| = 1, (1)u → 0 as |x| → ∞, are considered in this paper, where f ∈ C((1, ∞) × R+, R+) and R+ = [0, ∞).In recent years, similar problems have been discussed by several authors; see [1,2,3,4,5,6,7,8,9,10,11] and references therein
Similar problems have been discussed by several authors; see [1,2,3,4,5,6,7,8,9,10,11] and references therein
In the remainder of this section, we recall some facts on the fixed point index for completely continuous operators on a cone in the Banach space in order to prove our main results
Summary
In [11], by using the norm-type cone expansion and compression theorem, Stanczy proved that problem (1) has at least one positive radial solution under the following conditions:. The Scientific World Journal multiple positive radial solutions of (1) Our results cover both sub- and superlinear problems. In the remainder of this section, we recall some facts on the fixed point index for completely continuous operators on a cone in the Banach space in order to prove our main results.
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