Classification and evolution of bifurcation curves for a multiparameter p-Laplacian Dirichlet problem

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Abstract

We study the classification and evolution of bifurcation curves for the multiparameter p-Laplacian Dirichlet problem {(φp(u(x)))+λuq(k=1nakurk)1=0,1<x<1,q>0,0=r1<r2<<rn,n2,ak>0for k=1,2,,n,u(1)=u(1)=0,where p>1, φp(y)=|y|p2y, (φp(u)) is the one-dimensional p-Laplacian, and λ>0 is a bifurcation parameter, and q>0 is an evolution parameter. We give a classification of totally five qualitatively different bifurcation curves for different q>0. More precisely, we prove that, on the (λ,u)-plane, each bifurcation curve is either a monotone curve if q(0,p1][rn+p1,) or has exactly one turning point where the curve turns to the right if q(p1,rn+p1). Hence the problem has at most two positive solutions for each λ>0. We also show evolution of five bifurcation curves as q varies from 0+ to .

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 p-Laplacian Dirichlet problem {(φp(u(x)))+λuq(k=1nakurk)1=0,1<x<1,q>0,0=r1<r2<<rn,n2,ak>0for k=1,2,,n,u(1)=u(1)=0, where p>1, φp(y)=|y|p2y, (φp(u)) is the one-dimensional p-Laplacian, nonnegative constants r1,r2,,rn, and a1,a2,,an with n2 are given, λ>0 is a bifurcation parameter, and q>0 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 fC2[0,) satisfies m0=m=0 and g(u)f(u)/up1 is increasing–decreasing on (0,). Note that g is increasing–decreasing on (0,) means that there exists a finite positive number A such that g(u)>0 on (0,A) and g(u)<0 on (A,).

In Theorem 2.1 below, assuming one additional suitable condition (C1) on f, we prove that T(α) has exactly one critical point, a minimum, on (0,). 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 f(u)uq(k=1nakurk)1 with constants q>0, 0=r1<r2<<rn, n2, and ak>0 for k=1,2,,n. 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 p>1 . Consider (1.1). Then uf(u)/f(u) is nonincreasing on (0,).

Proof of Lemma 3.1

For (1.1), let h(u)f(u)/uq=(k=1nakurk)1, then uf(u)f(u)=quqh(u)+uq+1h(u)uqh(u)=q+uh(u)h(u). So uf(u)/f(u) 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 ak (1kn, n2) for f(u)=uq(k=1nakurk)1 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 f(u)=uq(k=1nakurk)1 with somenegative coefficient(s) ak, 1<k<n. For example, for (1.1), take p=2 and f(u)=u(1u+u2)1. It can be proved that f

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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This work is partially supported by the National Science Council of the Republic of China under grant No. 95-2115-M-007-013-MY3.

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