Abstract
We consider equations involving the onedimensional pLaplacian
with the Dirichlet boundary conditions. By using time map methods, we show how changes
of the sign of
MSC: 34B10, 34B18.
Keywords:
positive solutions; onedimensional pLaplacian; uniqueness; time map1 Introduction
Let
and
(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 [1], Peitgen et al.[2], Peitgen and Schmitt [3], Hess [4], Ambrosetti and Hess [5], Cosner and Schmitt [6], Dancer and Schmitt [7], Espinoza [8], Anuradha and Shivaji [9], Anuradha et al.[10], de Figueiredo [11], Lin and Pai [12], Clément and Sweers [13] and the references therein.
Very recently, Loc and Schmitt [14] considered the problem
where
then the problem (1.3) has at least
In the special case that
Theorem A [[1], Theorem 3.8]
Assume that
(H1)
(H2)
(H3) There exists
(H4) If
Then:
(a) For all
(b) If
where
and
(c) If
where
Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the onedimensional pLaplacian
We shall answer these questions in the affirmative if
Theorem 1.1Let
(H2′) either
Then:
(a) For all
(b) If
(c) If
where
We shall apply the time map method to show how changes of the sign of
In the following, we extend f so that
2 Preliminaries
To prove our main results, we use the uniqueness results due to Reichel and Walter [15] on the initial value problem
where
Lemma 2.1Let (H1) hold. If
Proof It is an immediate consequence of Reichel and Walter [[15], Theorem 2]. □
Lemma 2.2Let (H1) hold. Let
Then the initial value problem
has a unique local solution.
Proof (H1) implies that f is locally Lipschitzian. This together with the assumption
Lemma 2.3Let
with
Proof Since g is independent of t, both
By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on
Lemma 2.4Let
with
(a)
(b)
(c)
Proof (a) Suppose on the contrary that
However, this is impossible since
(b) Suppose on the contrary that there exists
We may assume that
If
This contradicts the boundary conditions
So, there exists
Obviously,
Multiplying both sides of the equation in (2.6) by
and subsequently,
This contradicts the facts that
Similarly, we can prove that
(c) Suppose on the contrary that there exists
This together with (2.8) implies that
This contradicts the facts that
3 Proof of the main results
To prove Theorem 1.1, we need the following preliminary results.
Lemma 3.1For any
has a positive solution
Proof By Lemma 2.4,
Suppose that
and so
Putting
Hence λ (if exists) is uniquely determined by ρ.
If
Let
Then
Lemma 3.2Let (H1) and (H2′) hold, and let
where
Proof We only deal with
To this end, we divide the proof into two cases.
Case 1. We show that
In this case, for any
for
where
see Zhang [16]. Hence
Case 2. We show that
where
In fact, (3.6) yields
for
We will show that the last integral in (3.9) converges to zeros as
For
For
uniformly in v. Therefore, (3.9) implies
Therefore, (3.7) holds. □
From the definitions of
Lemma 3.3Let
(i)
(ii)
Proof (i) Suppose firstly that
Clearly k must be a local maximum for F and so
Let
Then if
Hence
As
since
Suppose next that
Since
Thus
(ii) Let
Hence, if
Hence, if
where
□
Proof of Theorem 1.1 (a) follows from the continuity of
(b) follows from the continuity of
(c)
Hence
Now, if
and so
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
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).
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