On Positive Radial Solutions for a Class of Elliptic Equations

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).

In the present paper, we continue the study in [4]. Under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, we improve the above positive numbers and by using the fixed point index. Furthermore, we obtain several existence theorems on 2 The Scientific World Journal multiple positive radial solutions of (1). Our results cover both sub-and superlinear problems. It seems to be difficult to utilize the norm-type cone expansion and compression theorem to prove our results.
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. Please refer to [12][13][14] for more details.
Let be a real Banach space and a cone in . The following lemma is a well-known result of the fixed point index theory, which will play an important role in the proof of our main results.
The paper is organized as follows. In Section 2 we change problem (1) into a singular two-point boundary value problem and then investigate the existence and multiplicity of its positive solutions. And some examples are presented in Section 2. Several theorems on existence and multiplicity of positive radial solutions of problem (1) are established in Section 3.
The Scientific World Journal 3 To prove Lemma 2, we need the following lemmas.
(iv) From (14), (v) It is easy to verify that 0 is a cone in . It follows from (14) and (30) that The proof is completed.
We list some conditions as follows which will be useful in this section.
Proof. The proof is similar to that of Lemma 3.1 in [4], so we only sketch it. Under (H 1 ), is well defined and for every V ∈ , V is nonnegative and continuous on [0, 1]. Note the property of ( , ); it is easy to see that ( ) ⊂ . (H 1 ) and Lebesgue's dominated convergence theorem ensure the continuity of . Finally, by using Ascoli-Arzela theorem, we can prove that is completely continuous.
(vi) By (H 7 ), there exists > 0 such that For any V ∈ ∩ Ω , we have Without loss of generality, we can suppose that has no fixed point on ∩ Ω . Suppose that there exist V 1 ∈ ∩ Ω and 1 ⩾ 0 such that V 1 = V 1 + 1 1ℎ . Then 1 > 0 and Then * ⩾ 1 > 0 and V 1 ⩾ * 1ℎ . Since ℎ is a positive linear operator, we have Hence, by (52) we have which contradicts the definition of * . Thus according to the property of omitting a direction for fixed point index, we have ( , ∩ Ω , ) = 0.
Remark 16. Lan and Webb have studied BVP (73) in [21]. Their key conditions are