Research Article | Open Access
Exact Multiplicity of Solutions for a Class of Singular Generalized One-Dimensional -Laplacian Problem
We describe the existence of positive solutions for a class of singular generalized one-dimensional -Laplacian problem. By applying the related fixed point theory in cone, some new and general results on the existence of positive solutions to the singular generalized -Laplacian problem are obtained. Note that the nonlinear term involves the first-order derivative explicitly.
Recently, increasing attention is paid to question of positive solution for singular boundary value problems [1–11]. One notices that the singular boundary value problems for ordinary differential equation describe many phenomena in applied mathematics and physical science, which can be found in the theory of nonlinear diffusion generated by nonlinear sources and in the thermal ignition of gases [12–17]. Moreover, there are excellent results of nonsingular problem, see [18–21] and the references therein.
Moreover, the nonlocal -Laplacian problems for ordinary differential equation have been studied extensively. There are many papers dealing with the existence of positive solutions for the nonlocal -Laplacian boundary value problem, in which the nonlinear term is independent of the first-order derivative with different boundary conditions. For example, Ma et al.  established some existence results of positive solutions for the problem The main tool is the monotone iterative technique. By applications of fixed point index theory, Cheung and Ren  investigated two classes of quasilinear nonlocal problems: Sufficient conditions for the existence of twin positive solutions are established. In , Feng and Ge studied nonlocal problem for the one-dimensional -Laplacian They obtained sufficient conditions for the existence of at least three solutions to the above problem by using the Avery and Peterson fixed point theorem.
However, there are not many concerning the boundary value problems, in which a generalized -Laplacian equation with a nonlocal boundary conditions. The motivation for the present work stems from many recent investigations in [16, 23–25] and references therein. Our purpose of this paper is to establish some sufficient conditions for the existence of triple positive solutions to a class of a singular generalized one-dimensional -Laplacian problem where is -Laplacian operator; that is, , , , , and , , , , , , satisfy), satisfy , ,(), is positive and nondecreasing on ,() may be singular at and/or , and , where ,() is an -Carathédory function; that is, for each , the mapping is Lebesgue measurable on ; for a.e. , the mapping is continuous on .
Under the above assumptions, some new and general results on the existence of multiple positive solutions to singular one-dimensional -Laplacian are obtained. Our results develop some results of Cheung and Ren  and include and improve the main results of Feng and Ge  and Zhang .
The rest of the paper is organized as follows. In Section 2, we provide some background material from the theory of cones in the Banach spaces, which are useful later. Some lemmas and criteria for the existence of three positive solutions for one-dimensional -Laplacian problems are established in Section 3. Finally, we give an example to illustrate our main results.
Let and be nonnegative continuous convex functionals on a cone , a nonnegative continuous concave functional on , a nonnegative continuous functional on , and , , , and positive numbers. We define the following convex sets and a closed set
Now we state the fixed point theorem due to Avery and Peterson .
Lemma 1. Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on satisfying for , such that, for some positive numbers and , Suppose that is completely continuous and there are positive numbers , , and with such that(), for ,() for with ,() and for with . Then has at least three fixed points such that
Denote that .
Lemma 2. Assume that and hold. If , then the boundary value problem has a unique solution
Proof. The proof is similar to Lemma 3.6 in , we omit it.
Let be the Banach space with the norm , where . Define a cone by Let the assumptions () and () hold. Denote that Evidently, . We take where satisfies , is constant. It is not difficult to check that for a.e. , , for a.e. , and satisfies , . That is to say . Hence, .
Lemma 4. Assume that () holds. If , then where .
Proof. The proof is similar to Lemma 3.1 in , we omit it.
Let the nonnegative continuous convex functionals and , the nonnegative continuous concave functional , and the nonnegative continuous functional be defined on cone by
In the view of assumption (), that is, and (). Thus, we can define an operator by
Lemma 5. Assume that hold. Then is completely continuous.
Proof. For any , by the definition of , we see that , . From and Lemma 3, we have , implies that is concave on , , , and . So we can conclude that . In what follows, we prove that is compact. Let , there exists a set such that . For any , the assumptions imply that
We take the arguments to show that the operator is completely continuous. In view of Lemma 4 and the continuity of in and , assume that satisfy . Moreover, is an increasing and continuously function, and we obtain that
This means that the operator is continuous.
Since may be singular at and/or , choose two sequences , satisfying for any , such that and as , respectively. Define and an operator sequence by Clearly, is a piecewise continuous function, and the operator is well defined. Further, we can see that is completely continuous.
Let , and , , . We will prove that approaches uniformly on . From the absolute continuity of integral, we obtain where . For any and a.e. , one has that For and a.e. , we have the same result. It is easy to see that, for any and a.e. , there is . Similarly, we can obtain that, for any and a.e. , , , respectively, In view of the continuity of function , from the above arguments, we have That is to say, the sequence is uniformly an approximate on any bounded subset of . Applying the Arzela-Ascoli lemma to the operator , we can conclude that is relatively compact; that is, is completely continuous.
3. Main Results
We are ready to apply the Avery and Peterson fixed point theorem to the operator to give sufficient conditions for the existence of at least three positive solutions to the problem . For convenience, we introduce the following notation:
Theorem 6. Suppose hold and for a.e. . If there exist positive numbers , , and with such that the following conditions are satisfied:(), ,(), ,(), , then the problem has at least three positive solutions , , and satisfying
Proof. It is known that boundary value problem has a solution if and only if solves the operator equation . By the definition of operator , from Lemma 4 and the concavity of , the functionals defined above satisfy , for any . We also note that for , which suffices to show that the conditions of Lemma 1 hold with respect to . First of all, we show that if the assumption () is satisfied, then
In fact, for , there is . With Lemma 4, there is , and the assumption () implies that for a.e. . On the other hand, for , we obtain that , and so is concave on , and there is . By using of is positive and monotone increasing on a.e. , and so
Therefore, (31) is satisfied.
To check if the condition in Lemma 1 is fulfilled, we choose , where is defined by (16). From the hypothesis conditions of the theorem, we have Equation (33) implies that . That is, . So for , and hold. Hence, for , we see that and for a.e. . Hence by the assumption (), one has that for a.e. . By the definition of the functional , we obtain Therefore, we get for .
Next, we show that the condition in Lemma 1 holds. In fact, if with , then
Finally, we assert that the condition in Lemma 1 also holds. Since , there holds that . Assume that with . Then, by the assumption we get Thus, the conditions in Lemma 1 hold, and so the problem has at least three positive solutions , , and such that (30) holds.
Similar to the above arguments, we will discuss the problem Now, we replace the assumption with the following:
() , satisfy , .
The cone is defined by
Lemma 8. Assume that holds. If , then where and .
In what follows, we define integral operators and , , and :
Theorem 9. Suppose that , , , and hold and for a.e. . If there exist positive numbers , , , and with such that the following conditions are satisfied:,,,then the problem has at least three positive solutions , , and satisfying
From Theorem 6 or Theorem 9, we can see that, when assumptions like , , and are imposed appropriately on the nonlinear term , we also establish the existence results of an arbitrary odd number of positive solutions of the problem or . Here, we only present the form of Theorem 6.
Theorem 10. Suppose that hold and for a.e. . If there exist positive numbers , , , and , with such that the following conditions are satisfied:(), ,(), ,(), , then