In this work, we establish the existence and multiplicity results of positive solutions for one-dimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions.
MSC: 34B10, 34B18.
Keywords:mean curvature equations; positive solutions; existence; fixed point index
The prescribed mean curvature problems like
have attracted much attention in recent years, see [1-4] and the references therein. Since the problem is quasilinear non-uniformly elliptic, it is more difficult to study the existence of classical solutions. The greatest obstacle is the lack of gradient estimate, such kind of estimate does not hold in general and boundary gradient blow-up may occur. This leads to some new phenomena very different from those in semilinear problems. Many well-known results of semilinear problems have to be reconsidered for this quasilinear problem. Motivated by the search for solutions of the above problem, many authors (see [5-13]) studied the existence of (positive) solutions for one-dimensional prescribed mean curvature equations with Dirichlet boundary conditions
If , Bonheure et al., Habets and Omari , Kusahara and Usami , Pan and Xing [12,13] studied the existence of (positive) solutions of (1.2) by using the variational method, lower and upper solutions method and time mapping method, respectively.
However, to the best of our knowledge, the existence and multiplicity of positive solutions for (1.1) are relatively few by the fixed point index theory. In this paper, based on the fixed point index theory, we shall investigate the existence and multiplicity of positive solution of (1.1) when f is ϕ-superlinear and ϕ-sublinear at 0 and ∞, respectively, here .
For convenience, we introduce some notations
We will also need the function , and let . By a similar method in [, Lemma 2.8], it is not difficult to verify that .
In the rest of this paper, we shall study the existence of positive solutions of (1.3) by using the fixed point index theory to give a brief and clear proof for the existence of positive solutions of (1.1). More concretely, we shall prove the following.
Remark 1.1 The results of Theorem 1.1 and Corollary 1.2 are different with the case which is the classical Dirichlet boundary value problem (1.2). This phenomenon is a striking feature of problem (1.1), which is just the reason why we study the existence of positive solutions of problem (1.1). It is pointed out that in equation of (1.1), having replaced with , Theorem 1.1 and Corollary 1.2 also hold as well as all of the proofs with obvious changes.
Proof By a simple computation, it follows that and . So ϕ is an odd, increasing homeomorphism with . Moreover, from , we get that ϕ is convex up on . Notice that is also an odd, increasing homeomorphism with . It is easy to verify that is concave up on .
Proof By integrating, it follows that (2.1) has the unique solution given by
Lemma 2.3 ([, Lemma 4.1 and Lemma 4.2])
We next state the fixed point index theorem which will be used to prove our results.
Lemma 2.4 ([, Chapter 6])
From Lemma 2.2, problem (1.3) is equivalent to the fixed point problem
Let v be the solution of the problem
Then we have
This together with ϕ is an increasing homeomorphism implies that
3 Proof of the main results
Then Lemma 2.5 implies that
From Lemma 2.4, it follows that and . By using the additivity-excision property of the fixed point index , we have that
This together with Lemma 2.4 implies . By using the additivity-excision property of the fixed point index , we have
By Lemma 2.8, there exist
Proof of Corollary 1.2 It is easy to show by the result of Theorem 1.1(i) and (ii). □
Example 3.1 Let us consider the following problem:
Example 3.2 Let us consider the following problem:
Obviously, we divided the discussion into two cases as follows.
Remark 3.1 It is worth to point out that our results only partly generalize the results of Habet and Omari  and Pan , since is more general than , and due to the limitation of the fixed point index method.
The authors declare that they have no competing interests regarding the publication of this paper.
YL and RM completed the main study, carried out the results of this article and drafted the paper. HG checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
The authors are very grateful to the anonymous referee for his valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
Atkinson, FV, Peletier, LA, Serrin, J: Ground states for the prescribed mean curvature equation: the supercritical case. Nonlinear Diffusion Equations and Their Equilibrium States, I, pp. 51–74. Springer, New York (1988) (Berkeley, CA, 1986)
Cano-Casanova, S, López-Gómez, J, Takimoto, K: A quasilinear parabolic perturbation of the linear heat equation. J. Differ. Equ.. 252, 323–343 (2012). Publisher Full Text
Coffman, C, Ziemer, W: A prescribed mean curvature problem on domains without radial symmetry. SIAM J. Math. Anal.. 22(4), 982–990 (1991). Publisher Full Text
Pan, H, Xing, R: Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions. J. Differ. Equ.. 254, 1464–1499 (2013). Publisher Full Text
Bonheure, D, Habets, P, Obersnel, F, Omari, P: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ.. 243, 208–237 (2007). Publisher Full Text
Hai, DD, Schmitt, K, Shivaji, R: Positive solutions of quasilinear boundary value problems. J. Math. Anal. Appl.. 217, 672–686 (1998). Publisher Full Text
Habets, P, Omari, P: Multiple positive solutions of a one-dimensional prescribed mean curvature problem. Commun. Contemp. Math.. 9(5), 701–730 (2007). Publisher Full Text
Pan, H: One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Anal.. 70, 999–1010 (2009). Publisher Full Text
Pan, H, Xing, R: Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Nonlinear Anal.. 74, 1234–1260 (2011). Publisher Full Text
Wang, H: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl.. 281, 287–306 (2003). Publisher Full Text
Wang, H: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ.. 109, 1–7 (1994). Publisher Full Text