On Positive Radial Solutions for a Class of Elliptic Equations

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 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).

1. Introduction

The existence and multiplicity of positive radial solution for the following elliptic boundary value problem are considered in this paper, where and .

In recent years, similar problems have been discussed by several authors; see [1–11] and references therein. The usual approaches include variational method [2, 6, 7, 9], topological method [4, 10, 11], and sub- and supersolution method [3, 5].

In [11], by using the norm-type cone expansion and compression theorem, Stańczy proved that problem (1) has at least one positive radial solution under the following conditions:(B1)for any , there exists a function with such that (B2) there exists a set of positive measure such that (B3) there exists a function with such that

In a recent paper [4], replacing the conditions listed above by the weaker ones uniformly with respect to    for suitable positive numbers    and  , the authors proved that problem (1) still has at least one positive radial solution.

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 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 [12–14] 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.

Lemma 1 (see [12–14]). Let be a bounded open set in with , a completely continuous operator, where denotes the null element of . Assume that has no fixed point on .(i)(Homotopy invariance) If for all and , then the fixed point index ;(ii)(omitting a direction) if there exists an element such that for all and , then ;(iii)(cone expansion) if for , then ;(iv)(additivity) suppose is an open subset of with and for ; then (v)if , then has at least one fixed point in .

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.

2. Positive Solutions of Singular Two-Point Boundary Value Problems

Looking for radial solutions of (1), where , one can substitute thus reducing (1) to the following singular two-point boundary value problem, which is singular at 1: where

It is well known that the solution of (7) in is equivalent to the solution of the following Hammerstein integral equation in : where Green's function

Define an operator as follows: Then the solution of (9) in is equivalent to the fixed point of in .

Let be our Banach space with the norm for all , , and , where . It is easy to show that and are cones in . Let be the open ball of radius in . Define a set by

For , define an operator by

Then we have the following lemma.

Lemma 2. For any,(i) is a completely continuous positive linear operator, and the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue ;(ii);(iii)there exist , such that (iv)define a functional by for ; then for;(v)let then is a cone in and , where is defined by (14).

To prove Lemma 2, we need the following lemmas.

Lemma 3 (see [15]). Suppose that is a Banach space, are completely continuous operators, , and then is a completely continuous operator.

Lemma 4 (see [16, 17]). Suppose that is a completely continuous linear operator and. If there exist and a constant such that , then the spectral radius and has a postive eigenfunction corresponding to its first eigenvalue .

Proof of Lemma 2. (i) It follows from the definition of that
Hence, by Lebesgue’s dominated convergence theorem, it is easy to see that . Obviously, and is a linear operator; namely, is a positive linear operator. Next, we will show that is completely continuous. For any natural number ), let
Then is continuous and for all . Let
It is clear that is completely continuous. For any and , according to (18), (19), and the absolute continuity of integral, we have where . Therefore, by Lemma 3, is a completely continuous operator.
It is obvious that there exists such that . Thus there is such that and for all . Take such that and for all . Then for ,
So there exists a constant such that for all . From Lemma 4, we have that the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
(ii) To prove , we only need to show
In fact, for every , from for , we have so
Notice that, for , thus,
It follows from (24) and (26) that, for all ,
So (22) holds; thus, maps into .
(iii) Since is a positive eigenfunction of , we know from the maximum principle (see [18]) that for all . Note that for ; we have . This implies that and (see [18]). Define a function on by
Then it is easy to see that is continuous on and for all . So, there exist such that for all . Thus for all .
(iv) From (14), for all . So is well defined. For ,
(v) It is easy to verify that is a cone in . It follows from (14) and (30) that
So for all . The proof is completed.

Denote

We list some conditions as follows which will be useful in this section.() and for any there exists a function such that there exists a function such that there exists a function such that ; ;there exists a number such that ()there exists a function such that )there exist with for and such that

Lemma 5. Assume (H₁) holds. Then is a completely continuous operator.

Proof. The proof is similar to that of Lemma 3.1 in [4], so we only sketch it. Under (H₁), is well defined and for every , is nonnegative and continuous on . Note the property of ; it is easy to see that . () and Lebesgue's dominated convergence theorem ensure the continuity of . Finally, by using Ascoli-Arzela theorem, we can prove that is completely continuous.

Lemma 6. Assume (H₁) holds.(i)If (H₂) is satisfied, then for sufficiently small positive number ;(ii)if (H₃) is satisfied, then for sufficiently large positive number ;(iii)if (H₄) is satisfied, then for sufficiently small positive number ;(iv)if (H₅) is satisfied, then for sufficiently large positive number ;(v)if (H₆) is satisfied, then ;(vi)if (H₇) is satisfied, then for sufficiently small positive number ;(vii)if (H₈) are satisfied, then for sufficiently large positive number .

Proof. (i) By (H₂), there exists such that
Define for ; then is a bounded linear operator with and the spectral radial . For every , it follows from (40) that
So
If there exist and such that , then it is easy to see that . Thus and . By induction, we have , . Then and taking the supremum on gives . By the spectral radius formula, we have which is a contradiction. According to the homotopy property invariance of fixed point index, we have .
(ii) By (), there exist and such that
From (), there is such that for all . Hence,
Define for ; then is a bounded linear operator and . Let . Set
Next we prove that is bounded. For any , from (45), we have
Thus
Since is the first eigenvalue of and , the first eigenvalue of , . Therefore, the inverse operator exists and
It follows from that . Hence, we have from (48) that
That is, is bounded. Choose ; then for all and . By the homotopy invariance property of fixed point index, we have .
(iii)–(v) have been proved in [4], so we skip it.
(vi) By (), there exists such that
For any , we have
Without loss of generality, we can suppose that has no fixed point on . Suppose that there exist and such that . Then and . Let
Then and . Since 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 .
(vii) From (39), there exist and such that
Since is bounded on , there is a constant such that
Thus for all . Hence, by (38) we have for all . Let ; then is a finite constant. Take
Suppose that there exist and such that ; then
Hence,
On the other hand,
By the maximum principle, for all . By for , we have . Thus from (62) and (61), we have
This is a contradiction. So, by the property of omitting a direction for fixed point index, we have . The proof is completed.

Now, we are ready to state our main results of this section.

Theorem 7. Assume that (H₁), (H₂), (H₃), and (H₆) hold; then the singular BVP (7) has at least two positive solutions.

Proof. According to Lemma 6, we can choose sufficiently small positive number and sufficiently large positive number satisfying , , and . From and the additivity property of the fix point index, we obtain Hence has at least two fixed points, one in and another in . That is, the singular BVP (7) has at least two positive solutions. The proof is completed.

Theorem 8. If (H₁) and one of the following conditions are satisfied, then the singular BVP (7) has at least one positive solution.(i)(H₂) and (H₅) hold;(ii)(H₂) and (H₆) hold;(iii)(H₂) and (H₈) hold;(iv)(H₃) and (H₄) hold;(v)(H₃) and (H₆) hold;(vi)(H₃) and (H₇) hold.

Proof. By the properties of the fixed point index, we only need to choose suitable positive numbers and . This completes the proof.

Remark 9. The following conditions are little stronger than (), (), and (), respectively. And they are somewhat easy to verify. In fact, () is a key condition in [19].() and there exist and such that for all ;();().
If () holds, then, for any , let ; then we have for all and . Consequently, () holds. From , it is easy to see that if () or () holds, then () or () holds, respectively.

Remark 10. In [11], Stańczy established a one-solution theorem for the singular problem (7). In addition to (), the key conditions imposed on the nonlinear term are the following ones:(A2)there exists a set of positive measures such that (A3)there exists a function such that
It is easy to see that (A2) is equivalent to the following condition:()there exists a set such that
In a recent paper [4], Han and Wang improved the results in [11] by substituting the following conditions for the above ones: (HW5) ; (HW2) there exists a function such that where for .
Additionally, they obtained a twin-solution theorem for the singular BVP (7) by using (HW2), (), and (HW3) there exists a function such that
Since , () is an improvement of (HW2). In fact, without loss of generality, suppose , the positive eigenfunction corresponding to , satisfies ; then
So and then (HW2) implies (). Therefore, Theorems 7 and 8 are essential improvements of Theorems 3.1, 3.2 in [4] and Theorem 2.2 in [11]. Furthermore, we give the new conditions () and (). These improvements allow us to deal with more singular problems.

Remark 11. For the following singular BVP with general boundary conditions, where , and and , our results still hold. In fact, the Green function has analogous properties as is defined in (10) (see [4, Remark 3.3]). Hence, Theorems 7, 8 can be generalized to the singular problem (71) without any essential difficulty. See [4, 19, 20] for details.

For the following singular two-point BVP whose variables of nonlinear term are separated, the hypotheses and results will be more concise:

Since is a fixed function, , , and are confirmed exclusively. So we skip the subscript in the following. Corresponding  to ()–(), we formulate the conditions for singular BVP (73):() and ;(); ();() and ;() and ;()there exists a number such that ();().

Remark 12. Take with . Then
Notice that , and ; we have
So () implies () and () implies ().

Remark 13. Observe that the condition is not contained in (). In fact, the cone will be replaced by the cone which is defined in (15) as we consider one solution by (). See the proof of Theorem 15 in the following.

Theorem 14. Assume that (), (), (), and () hold; then the singular BVP (73) has at least two positive solutions.

Proof. This theorem is a direct corollary of Theorem 7.

Theorem 15. If ()  and one of the following conditions are satisfied, then the singular BVP (7) has at least one positive solution.(i)() and () hold;(ii)() and () hold;(iii)() and () hold;(iv)() and () hold;(v)() and () hold;(vi)() and () hold.

Proof. (i), (ii), and (iv)–(vi) are direct corollaries of Theorem 7. Next, we prove (iii). In contrast to (), () does not contain the condition that for .
(iii) Since   is completely continuous, we know that is also completely continuous. Similar to item (i) of Lemma 6, by (), it is not difficult to prove for sufficiently small positive number . By (), there exist and such that
Since is bounded on , there is a constant such that for all . Thus
Let ; then is a finite constant. Take
Suppose that there exist and such that ; then from (79) we have
Hence,
On the other hand, since , we have
Thus from (82) and (83), we have
This contradicts with (80). By the property of omitting a direction for fixed point index, we have . So . Thus has a fixed point in ; that is, the singular BVP (73) has at least one positive solution. The proof is completed.

Remark 16. Lan and Webb have studied BVP (73) in [21]. Their key conditions are(i), a.e. on and ;(ii)One of the following conditions holds:(h1) and ;(h2) and , where and .
Under the above condition (i), we still can prove Lemma 2 (see Theorem 2.1 in [21]). By (70) and (75), we have . Therefore, the items (iii) and (vi) of Theorem 15 extend the main results in [21] essentially. Furthermore, our conditions (), (), (), and () cannot be improved anymore.

At the end of this section, we present three simple examples to which our theorems can be applied, respectively. We choose .

Example 17. Let where is a constant. Obviously, for all , where and
Since , if and , then satisfies all the conditions of Theorem 7; thus, we infer that the singular BVP (7) has at least two positive solutions. Furthermore, if then Theorem 3.1 in [4] is invalid for this example.

Example 18. Let where . Since and , the item (iv) of Theorem 8 implies that singular BVP (7) has at least one positive solution.

Example 19. Let where , for , and . Since and , the item (iii) of Theorem 15 ensures that the singular BVP (73) has at least one positive solution.

3. Positive Radial Solutions of Elliptic Boundary Value Problems

Define a set

Denote and . For , let

Then we have . As in (13) and Lemma 2, confirms an operator and its first eigenvalue . To emphasize their relation with , we use the notations , , and .

According to (8), we formulate the following conditions which correspond to those in Section 2.() and for any there exists a function such that ()there exists a function such that ()there exists a function such that ();();()there exists a number such that ()there exists a function such that ()there exist with for and such that

Now we are ready to state our main results for the elliptic BVP (1).

Theorem 20. If (), (), (), and () are satisfied, then the elliptic BVP (1) has at least two positive radial solutions.

Theorem 21. If () and one of the following conditions are satisfied, then the elliptic BVP (1) has at least one positive radial solution:(i)() and () hold;(ii)() and () hold;(iii)() and () hold;(iv)() and () hold;(v)() and () hold;(vi)() and () hold.

Remark 22. According to Remark 10, we know
Therefore, Theorems 20 and 21 are essential improvements of Theorems 4.1 and 4.2 in [4], respectively.

Remark 23. For the following BVP whose nonlinear terms are variable-separated, it is easy to give more concise theorems as Theorems 14 and 15; we skip them here: where and .

Remark 24. One can see that, through variables substitution, it is easy to give some examples like in Section 3, so we skip it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are supported by the National Natural Science Foundation of China (11101253 and 10826081), the Fundamental Research Funds for the Central Universities (GK200902046 and GK201301007), and the Scientific Research Foundation of Xi’an University of Science and Technology (no. 200843).