Keywords:positive solutions; one-dimensional p-Laplacian; uniqueness; time map
(and their finite difference analogues) have been extensively studied since early 1980s. Several different approaches such as variational methods, bifurcation theory, lower and upper solutions method and quadrature arguments have been successfully applied to show the existence of multiple solutions. See Brown and Budin , Peitgen et al., Peitgen and Schmitt , Hess , Ambrosetti and Hess , Cosner and Schmitt , Dancer and Schmitt , Espinoza , Anuradha and Shivaji , Anuradha et al., de Figueiredo , Lin and Pai , Clément and Sweers  and the references therein.
Very recently, Loc and Schmitt  considered the problem
where is the p-Laplace operator for . They assumed that the nonlinearity f is a continuous function on ℝ, , and there exist such that on and on for every . They proved that, for λ sufficiently large, if
In the special case that and , Brown and Budin  applied the quadrature arguments to get the following more detailed results.
Theorem A [, Theorem 3.8]
Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the one-dimensional p-Laplacian
To prove our main results, we use the uniqueness results due to Reichel and Walter  on the initial value problem
Proof It is an immediate consequence of Reichel and Walter [, Theorem 2]. □
Then the initial value problem
has a unique local solution.
Proof (H1) implies that f is locally Lipschitzian. This together with the assumption and using [, (iii) and (v) in the case (β) of Theorem 4] yields that (2.2) has a unique solution in some neighborhood of a. □
Similarly, we can prove that
This together with (2.8) implies that
3 Proof of the main results
To prove Theorem 1.1, we need the following preliminary results.
Hence λ (if exists) is uniquely determined by ρ.
If , we define by (3.6) and by (3.5). It is straightforward to verify that u is twice differentiable, u satisfies (3.3), (3.4), in and . The continuity of is implied by (3.6) and this completes the proof. □
To this end, we divide the proof into two cases.
see Zhang . Hence
In fact, (3.6) yields
uniformly in v. Therefore, (3.9) implies
Therefore, (3.7) holds. □
The authors declare that they have no competing interests.
RM completed the main study, carried out the results of this article. YL drafted the manuscript. AOMA checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), SRFDP (No. 20126203110004) and Gansu Provincial National Science Foundation of China (No. 1208RJZA258).
Brown, KJ, Budin, H: On the existence of positive solutions for a class of semilinear elliptic boundary value problems. SIAM J. Math. Anal.. 10(5), 875–883 (1979). Publisher Full Text
Peitgen, HO, Saupe, D, Schmitt, K: Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions. J. Reine Angew. Math.. 322, 74–117 (1981)
Hess, P: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Commun. Partial Differ. Equ.. 6(8), 951–961 (1981). Publisher Full Text
Ambrosetti, A, Hess, P: Positive solutions of asymptotically linear elliptic eigenvalue problems. J. Math. Anal. Appl.. 73(2), 411–422 (1980). Publisher Full Text
Cosner, C, Schmitt, K: A priori bounds for positive solutions of a semilinear elliptic equation. Proc. Am. Math. Soc.. 95(1), 47–50 (1985). Publisher Full Text
Dancer, EN, Schmitt, K: On positive solutions of semilinear elliptic equations. Proc. Am. Math. Soc.. 101(3), 445–452 (1987). Publisher Full Text
Anuradha, V, Shivaji, R: Existence of infinitely many nontrivial bifurcation points. Results Math.. 22(3-4), 641–650 (1992). Publisher Full Text
de Figueiredo, DG: On the existence of multiple ordered solutions of nonlinear eigenvalue problems. Nonlinear Anal.. 11(4), 481–492 (1987). Publisher Full Text
Lin, S-S, Pai, FM: Existence and multiplicity of positive radial solutions for semilinear elliptic equations in annular domains. SIAM J. Math. Anal.. 22(6), 1500–1515 (1991). Publisher Full Text
Zhang, M: Nonuniform nonresonance of semilinear differential equations. J. Differ. Equ.. 166(1), 33–50 (2000). Publisher Full Text