Multiplicity of radially symmetric solutions for a p-harmonic equation in
© Li and Hui; licensee Springer. 2013
Received: 24 February 2013
Accepted: 15 November 2013
Published: 17 December 2013
This paper is concerned with the multiplicity of radially symmetric positive solutions of the Dirichlet boundary value problem for the following N-dimensional p-harmonic equation of the form
where is a unit ball in (). We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of radially symmetric positive solutions. We find that there exists a threshold such that if , the problem has no radially symmetric positive solution; while if , the problem admits at least one radially symmetric positive solution. Especially, there exist at least two radially symmetric positive solutions for .
In this paper, we discuss the multiplicity of positive radially symmetric solutions for the problem (P), namely, problem (1.1)-(1.2). The main purpose of this paper is to investigate the existence, nonexistence and multiplicity of positive solutions of problem (1.1)-(1.2). Different from some known works, the equation we consider is quasilinear and it might have degeneracy or singularity. In fact, if , the equation is degenerate at the points where ; while if , then the equation has singularity at the points where . And for the restrictions of the boundary value conditions, the above-mentioned operator Φ in (1.3), which is suitable for the entire space, is inappropriate in this paper. Hence, we propose a new analogue of the operator Φ and apply the fixed point index theory combining with the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions for the problem (P).
(H1) is continuous and nondecreasing on . Furthermore, there exist and such that , ;
(H2) is continuous, , and on any subinterval of .
The main result of this paper is the following.
Theorem 1 Let (H1) and (H2) hold true. Then there exists a threshold such that problem (1.1)-(1.2) has no positive solution for , has at least one positive solution for , and especially has at least two positive solutions for .
This paper is organized as follows. As preliminaries, we state some necessary lemmas in Section 2. In the last section, we apply the fixed point index theory and the upper and lower solutions method to prove the main result.
In this section, we first present the necessary definitions and introduce some auxiliary lemmas, including those from the fixed point index theory and the theory based on the upper and lower solutions method.
Definition 1 Let . We say that u is a positive solution of problem (1.1)-(1.2) if it satisfies (1.1)-(1.2) and on .
where is an operator defined on the Banach space E.
If we change ‘≤’ in the above inequality by ‘≥’, we can obtain the definition of a lower solution.
We say that u is a solution of problem (2.1), if u is an upper solution and is also a lower solution.
Throughout this paper, the main tools, which can be used to obtain the multiplicity of solutions for the problem, are the following two lemmas related to the fixed point index theory and the theory based on upper and lower solutions method, respectively, see .
respectively, then the above equation has a minimum solution and a maximum solution in the ordered interval , and .
We also need the following technical lemma on the property of the function f, see .
It is easy to prove that has the following properties.
So there exists such that , where satisfies . Hence, and . The proof is complete. □
By (2.10), it is easy to obtain the following lemma, which is proved by a direct computation.
Lemma 4 Let (H1) and (H2) hold true. Then problem (1.1)-(1.2) has a solution u if and only if u is a fixed point of . And equations (2.8)-(2.9) have a solution u if and only if u is a fixed point of .
Now, we discuss the properties of the function for .
Proof First, we want to prove that for any , there exists such that and for .
implies that for any , there exists such that and for .
By the front argumentation, has a null point and for . If , then there must exist a subsequence of such that and . Due to , by the continuity of , we then have , which contradicts the fact that . □
where θ is defined in Lemma 6. It is clear that the nonnegative continuous concave functions are in K.
In order to apply the fixed point index theory, the following two lemmas, which relate to the monotonicity and the continuity of the operator , are necessary. The proofs of Lemma 7 can be obtained immediately, by using (H1) and (2.10) with some simple direct computations.
Lemma 7 Let (H1) and (H2) hold true. Then the operator defined by (2.10) is a monotonic increasing operator, i.e., if , then , where `≤’ is the partial order defined on K.
Lemma 8 Let (H1) and (H2) hold true. Then the operator is completely continuous, and .
Therefore, we see that is continuous. And the compactness of the operator is easily obtained from the Arzela-Ascoli theorem.
which implies . Hence, we obtain . □
Now, we give the a priori estimates on the positive solutions of problem (1.1)-(1.2).
Lemma 9 Let (H1) and (H2) hold true. And suppose that , . Then there exists such that , where , and is a solution of problem (1.1)-(1.2) with instead of λ.
Therefore, . □
3 Existence of positive solutions
In this section, we give the proof of the main result, that is, Theorem 1. The proof will be divided into two parts. Firstly, by the upper and lower solutions method, we investigate the basic existence of positive solutions of problem (1.1)-(1.2). Exactly, we will determine the threshold of the parameter λ such that the problem is solvable if and only if . Finally, by utilizing the fixed point index theory, we establish the multiplicity of positive solutions for the case .
We first present and prove the basic existence result of positive solutions of problem (1.1)-(1.2).
Proposition 2 Let (H1) and (H2) hold true. Then there exists with such that problem (1.1)-(1.2) admits at least one positive solution for and has no positive solution for any .
which implies that is an upper solution of . It is obvious that, for all , is a lower solution of , and , . Hence, , where is the ordered interval in E. By Lemma 2, has a fixed point for . Therefore is a solution of problem (1.1)-(1.2). And then, for any , we have , which implies that .
which implies that is an upper solution of . Combining this with the fact that for , is a lower solution of , and therefore, by Lemma 2, Lemma 4, Lemma 7 and Lemma 8, problem (1.1)-(1.2) has a solution, therefore , which implies that .
Letting in (3.1)-(3.2), we can obtain the contradiction. Therefore we have .
It remains to show that . Let, (), be an increasing sequence. Suppose that is the solution of (1.1)-(1.2) with instead of λ. By Lemma 9, there exists such that , . Therefore, is an equicontinuous and bounded uniformly subset in . By the Ascoli-Arzela theorem, has a convergent subsequence. Without loss of generality, we suppose (). Since , due to the continuity of , we see that is continuous and bounded uniformly in , from which together with the continuity of and (H2), by the Lebesgue dominated convergence theorem, we have . Hence, by Lemma 4, is a solution of problem (1.1)-(1.2) with instead of λ. The proof is complete. □
Finally, we prove the main result in this paper.
Proof of Theorem 1 The arguments are based on fixed point index theory. Exactly speaking, we apply Lemma 1 to calculate the indexes of the corresponding operator in different two domains, and then complete the proof by the index theory.
To calculate the index of the operator on some subset of K, we need to check the validity of the conditions in Lemma 1.
Since , . Therefore, there is a fixed point of in Ω and a fixed point of in , respectively. Finally, by utilizing Lemma 4, it follows that (1.1)-(1.2) has at least two positive solutions for the case , which combined with Proposition 2 yields Theorem 1. □
This work is supported by the National Natural Science Foundation of China (Grant No. 71303067), China Postdoctoral Science Foundation funded project (Grant No. 2013M541400), the Heilongjiang Postdoctoral Fund (Grant No. LBH-Z12102), and the Fundamental Research Funds for the Central Universities (Grant No. HIT.HSS.201201).
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