Journal of Applied Mathematics

Volume 2014, Article ID 810193, 9 pages

http://dx.doi.org/10.1155/2014/810193

## Existence of Sign-Changing Solutions to Equations Involving the One-Dimensional -Laplacian

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 23 March 2014; Accepted 21 July 2014; Published 17 August 2014

Academic Editor: Mehmet Sezer

Copyright © 2014 Ruyun Ma and Lingfang Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the equations involving the one-dimensional -Laplacian , and where and . We show the existence of sign-changing solutions under the assumptions and . We also show that has exactly one solution having specified nodal properties for for some . Our main results are based on quadrature method.

#### 1. Introduction and Main Results

Existence and multiplicity of positive solutions of nonlinear second order boundary value problem and its generalized forms have been extensively studied via the fixed point theorem in cones, bifurcation theory, quadrature method, and fixed index theorem in the past four decades; see Erbe and Wang [1], Henderson and Wang [2], Laetsch [3], Fink et al. [4], Ma and Thompson [5, 6], and the references therein.

Existence and multiplicity of positive solutions of the corresponding one-dimensional -Laplacian have also been studied by several authors; see Lee and Sim [7], Wang [8], Kong and Wang [9], Aranda and Godoy [10], Bouguima and Lakmeche [11], and de Coster [12] for references along this line.

Recently, Lee and Sim [7] consider the existence and multiplicity of positive solutions of (2), (3) under the assumptions They proved the following.

Theorem A (see [7, Theorem 3.14]). *Assume (4) hold. Then, there exist such that (2), (3) have at least one positive solution for and no positive solution for .*

Of course, natural question is as follows. What would happen if we allow that ?

It is the purpose of this paper to study sign-changing solutions of (2), (3) under the assumptions and or The main tool is the quadrature method.

We will make the following assumptions: (H0) for ; (H1); (H2) and .

Let , where . The main results of this paper are the following.

Theorem 1. *Let (H0), (H1), and (H2) hold. Assume that satisfies
**
Then, for , (2), (3) have two solutions and for each : has zeros in and is positive near , and has zeros in and is negative near . Moreover, there exists a constant , such that for each the above solution is unique.*

Theorem 2. *Let (H0), (H1), and (H2) hold. Assume that . Then, for , (2), (3) have two solutions and for each : has zeros in and is positive near , and has zeros in and is negative near . Further, there exists a constant independent of , such that for each the above solution is unique.*

Theorem 3. *Let (H0) and (H1) hold. Assume that . Then, for , there exists a constant small independent of , such that for each (2), (3) have two solutions and : and have zeros in and are positive near 0; and problems (2), (3) have two solutions and , where and have zeros in and are negative near .*

*Remark 4. *For , the existence of positive and sign-changing solutions has been extensively studied by many authors [1–6], but they did not give any information about the uniqueness of nodal solutions.

*Remark 5. *It is worth noticing that Lee and Sim [7] studied the nonautonomous cases (2), (3) and obtained the existence of positive solutions with . They gave no information about the sign-changing solutions. In Theorem 1, we show the existence of solutions having specified nodal properties.

*Remark 6. *Very little is known in the available literature even in the special case . We establish uniqueness results in this paper; see Theorems 1 and 2.

*Remark 7. *Let us consider the problem
where . Obviously, satisfies (H0) and (H1). Since
it is easy to see that (H2) is fulfilled. Thus, Theorem 1 implies that, for , (6) have two solutions and for each : has zeros in and is positive near , and has zeros in and is negative near . Moreover, there exists a constant , such that for each the above solution is unique.

For other results dealing with -Laplacian operators and the bifurcation behavior of solutions, see [13–24] and the references therein.

The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results. Finally, in Section 3, we give the proofs of Theorems 1, 2, and 3.

#### 2. Quadrature Method and Preliminaries

Let for and .

Lemma 8. *If is any solution of (2), (3) and is such that , then .*

*Proof. *Since is autonomous, both and satisfy the initial value problem
By Reichel and Walter [14, Theorem 2] and [14, (iii) and (v) in the case () of Theorem 4], (8) has a unique solution defined on . Therefore, .

Now, we divide the discussion into two cases.

*Case 1 (.). *In this case, we attempt to find a solution of (2), (3) with zeros in (0, 1) and and a solution of (2), (3) with zeros in (0, 1) and .

Obviously, if is a sign-changing solution with zeros in and , then, thanks to Lemma 8 and the fact that (2) is autonomous, we only need to study on the intervals and .

Multiplying (2) throughout by , we obtain and integrating we have If and , then and . Substituting and in (10), we get and . Hence, is such that Thus, we have Integrating (12) and (13) on and , respectively, we obtain Hence, substituting in (14) and in (15), we have Multiplying (17) by and adding to (16), we can see that and satisfy In fact, the following result holds.

Lemma 9. *Given , if there exists such that , then (2), (3) have a sign-changing solution with interior zeros satisfying . Further, is a continuous function in and it is also differentiable with the derivative given by
**
where .*

The proof of the above theorem follows by carefully extending the arguments used in [15, Theorem 2.2] for second order differential equation to the case of one-dimensional -Laplacian.

Using the same argument, with obvious changes, we may deduce the following.

If is a sign-changing solution with zeros in and , the corresponding is

*Case 2 (). *In this case, if is a sign-changing solution with zeros in and , the corresponding is
Similarly, we may get the same function as above when is a sign-changing solution with zeros in with .

#### 3. The Proofs of the Main Results

*Proof of Theorem 1. *First, we consider .

It follows from the quadrature method that a solution with zeros in (0,1) exists if for there exists such that . To prove this, we will show that . We achieve this by proving (A),
(B). *Proof of (A)*. Recall that

First let us consider

To this end, we have from (H1) that, for any , there exists , such that

If , it follows from (23) and (24) that we have that
where
It follows from the fact that is sufficiently large and (25) that

Next, we know that as (). We consider
To this end, we have from (H1) that, for any , there exists , such that

If , it follows from (28) and (29) that we have that
where
It follows from the fact that is sufficiently large and (30) that
Therefore, from (27) and (32), we have that .*Proof of (B)*. Recall that

First, let us consider

Since , then, for any , there exists such that
Thus, if , the second part of (35) implies that
Similarly, from the first part of (35), we have that
It follows from (36), (37) and the fact is arbitrary that
In fact, as (); we consider
From , then, for any , there exists such that
Thus, if , the second part of (40) implies that
Similarly, from the first part of (40), we have that
It follows from (41), (42) and the fact that is arbitrary that we have that

Therefore, from (40) and (43), we have that
By analysing defined in (20) instead of in the proof of the above, we have the same result. Thus, we have shown that there are two solutions with interior zeros, which are negative near 0 and positive near 0 for , respectively.

Now, in order to achieve the existence of , we will first establish that for large enough. In fact,
First, we consider
where . From the first part of (H2), it follows that
if is large enough.

Next, let us consider
where . From the second part of (H2), we have that for and large enough. In fact, as . Consequently, we get that for large enough.

Finally, if , this clearly follows by analysing defined in (21) instead of in the proof of the case .

*Proof of Theorem 2. *First, we consider .

It follows from the quadrature method that a solution with interior zeros exists if for there exists such that . To prove this, we will show that . We achieve this by proving (A1), (B1). The proof of (A1) is the same as the proof of (A) of Theorem 1, so we omit it here; we are only to prove (B1). Recall that

First let us consider

From , then, for any , there exists such that
Thus, if , from (50), we have that

Next, in fact, as ; we consider

Since , then, for any , there exists such that
Thus, if , from (54), we have that
From the fact that is small and combining (52) and (55), we get that
By analyzing defined in (20) instead of in the proof of the above, we have the same result. The proof of is similar to the proof of Theorem 1. We omit it here.

Finally, if , then it clearly follows by analyzing defined in (21) instead of in the proof of the case .

*Proof of Theorem 3. *First, we consider .

It follows from the quadrature method that a solution with interior zeros exists if for there exists such that . We prove this by proving (A2), (B2).The proof of (A2) is the same as the proof of (A) of Theorem 1, so we omit it here; we are only to prove (B2).

Recall that

First, consider

Since , then, for any large , there exists such that
Thus, if , from (58) and (59), we have that

Next, in fact, as (); we consider

Since , then, for any large , there exists such that

Thus, if , from (62), we have that
From the fact that is arbitrary and large and combining (60) and (63), we get that
By analyzing defined in (20) instead of in the proof of the above, we have the same result.

Finally, if , then it clearly follows by analyzing defined in (20) instead of in the proof of the case .

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the NSFC (no. 11361054), SRFDP (no. 20126203110004), and Gansu Provincial National Science Foundation of China (no. 1208RJZA258).

#### References

- L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations,”
*Proceedings of the American Mathematical Society*, vol. 120, no. 3, pp. 743–748, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Henderson and H. Wang, “Positive solutions for nonlinear eigenvalue problems,”
*Journal of Mathematical Analysis and Applications*, vol. 208, no. 1, pp. 252–259, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Laetsch, “The number of solutions of a nonlinear two point boundary value problem,”
*Indiana University Mathematics Journal*, vol. 20, pp. 1–13, 1971. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. Fink, J. A. Gatica, and G. E. Hernández, “Eigenvalues of generalized Gel'fand models,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 20, no. 12, pp. 1453–1468, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Ma and B. Thompson, “Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities,”
*Journal of Mathematical Analysis and Applications*, vol. 303, no. 2, pp. 726–735, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. Ma, “Nodal solutions of second-order boundary value problems with superlinear or sublinear nonlinearities,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 4, pp. 950–961, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Lee and I. Sim, “Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,”
*Journal of Differential Equations*, vol. 229, no. 1, pp. 229–256, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Wang, “The existence of positive solutions for the one-dimensional $p$-Laplacian,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 8, pp. 2275–2283, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Kong and J. Wang, “Multiple positive solutions for the one-dimensional $p$-Laplacian,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 42, no. 8, pp. 1327–1333, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Aranda and T. Godoy, “Existence and multiplicity of positive solutions for a singular problem associated to the $p$-Laplacian operator,”
*Electronic Journal of Differential Equations*, vol. 2004, no. 132, 15 pages, 2004. View at Google Scholar · View at MathSciNet - S. M. Bouguima and A. Lakmeche, “Multiple solutions of a nonlinear problem involving the $p$-Laplacian,”
*Communications on Applied Nonlinear Analysis*, vol. 7, no. 3, pp. 83–96, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. de Coster, “Pairs of positive solutions for the one-dimensional $p$-Laplacian,”
*Nonlinear Analysis*, vol. 23, no. 5, pp. 669–681, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Lee and I. Sim, “Existence results of sign-changing solutions for singular one-dimensional $p$-Laplacian problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 5, pp. 1195–1209, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. Reichel and W. Walter, “Radial solutions of equations and inequalities involving the $p$-Laplacian,”
*Journal of Inequalities and Applications*, vol. 1, no. 1, pp. 47–71, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Anuradha and R. Shivaji, “Existence of infinitely many nontrivial bifurcation points,”
*Results in Mathematics*, vol. 22, no. 3-4, pp. 641–650, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - L. Gasińki and N. S. Papageorgiou, “Bifurcation-type results for nonlinear parametric elliptic equations,”
*Proceedings of the Royal Society of Edinburgh A: Mathematics*, vol. 142, no. 3, pp. 595–623, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Gasiński, “Positive solutions for resonant boundary value problems with the scalar $p$-Laplacian and nonsmooth potential,”
*Discrete and Continuous Dynamical Systems A*, vol. 17, no. 1, pp. 143–158, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Filippakis, L. Gasiski, and N. S. Papageorgiou, “Multiple positive solutions for eigenvalue problems of hemivariational inequalities,”
*Positivity*, vol. 10, no. 3, pp. 491–515, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Castro and R. Shivaji, “Nonnegative solutions for a class of nonpositone problems,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 108, no. 3-4, pp. 291–302, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 203, no. 3, pp. 610–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - P. Drabek, P. Girg, P. Takac, and M. Ulm, “The Fredholm alternative for the $p$-Laplacian: bifurcation from infinity, existence and multiplicity,”
*Indiana University Mathematics Journal*, vol. 53, no. 2, pp. 433–482, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. J. Brown, M. M. A. Ibrahim, and R. Shivaji, “$S$-shaped bifurcation curves,”
*Nonlinear Analysis*, vol. 5, no. 5, pp. 475–486, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Manásevich and F. Zanolin, “Time-mappings and multiplicity of solutions for the one-dimensional $p$-Laplacian,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 21, no. 4, pp. 269–291, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - I. Addou, S. M. Bouguima, M. Derhab, and Y. Raffed, “Multiple solutions of Dirichlet problems,”
*Dynamic Systems an Applications*, vol. 7, no. 4, pp. 575–601, 1998. View at Google Scholar