Classification and evolution of bifurcation curves for a multiparameter -Laplacian Dirichlet problem☆
Section snippets
Introduction and main result
In this paper we mainly study the classification and evolution of bifurcation curves of positive solutions for the multiparameter -Laplacian Dirichlet problem where , , is the one-dimensional -Laplacian, nonnegative constants , and with are given, is a bifurcation parameter, and is an evolution parameter.
An interesting example of the
Statements and proofs of Theorem 2.1 and Corollary 2.2
To (1.4), we consider satisfies and is increasing–decreasing on . Note that is increasing–decreasing on means that there exists a finite positive number such that on and on .
In Theorem 2.1 below, assuming one additional suitable condition (C1) on , we prove that has exactly one critical point, a minimum, on . In Corollary 2.2, we give a sequence of simplified sufficient conditions for (C1), which are easier to
Proof of Theorem 1.1
To (1.4), in this section, let with constants , , , and for . Then (1.4) reduces to (1.1), and we then apply time mapping results obtained in Sections 1 Introduction and main result, 2 Statements and proofs of to prove Theorem 1.1. We need one more lemma.
Lemma 3.1 Let . Consider (1.1). Then is nonincreasing on .
Proof of Lemma 3.1 For (1.1), let , then So is
A remark to Theorem 1.1 and two applications of Corollary 2.2
We first give next remark to Theorem 1.1.
Remark 4 The positivity assumption of all coefficients (, ) for in (1.1) may be weakened. Our time-map techniques as in the proof of Theorem 1.1(iii) can be suitably modified such that we are able to obtain the same exact multiplicity result in Theorem 1.1(iii) for some nonlinearities with somenegative coefficient(s) , . For example, for (1.1), take and . It can be proved that
Acknowledgements
The authors thank Prof. Junping Shi for many illuminating discussions and suggestions. The authors also thank the referee for a careful reading of the manuscript and valuable suggestions on the manuscript. Much of the computation in this paper has been checked using the symbolic manipulator Mathematica 7.0.
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Exact multiplicity of positive solutions for a p-Laplacian equation with positive convex nonlinearity
2016, Journal of Differential EquationsCitation Excerpt :An approach based on maximum principle, linearized equations and implicit function theorem was used in [28,29] to prove the nondegeneracy and uniqueness of positive solution of (Pλ) on a ball. For the one-dimensional problem, the equation (Pλ) can be integrated via a quadrature method and the uniqueness or exact multiplicity of positive solutions of (Pλ) can be proved by analyzing the associated time-map, see [30–34]. Very recently, the exact multiplicity of positive solutions of (Pλ) on an annulus was showed in [35] by using the Kolodner–Coffman method.
Families of Solution Curves for Some Non-autonomous Problems
2016, Acta Applicandae MathematicaeExact multiplicity of sign-changing solutions for a class of second-order dirichlet boundary value problem with weight function
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This work is partially supported by the National Science Council of the Republic of China under grant No. 95-2115-M-007-013-MY3.