Global Bifurcation Techniques Research Articles

Global bifurcation for a class of nonlinear ODEs

We briefly survey global bifurcation techniques, and illustrate their use by finding multiple positive periodic solutions to a class of second order quasilinear ODEs related to the Yamabe problem. As an application, we give a bifurcation-theoretic proof of a classical nonuniqueness result for conformal metrics with constant scalar curvature, that was independently discovered by Kobayashi and Schoen in the 1980s.

Global bifurcation of solitary waves to the Boussinesq abcd system

The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a,b,c,d) satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b≠d). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.

Continuum of one‐sign solutions of one‐dimensional Minkowski‐curvature problem with nonlinear boundary conditions

In this work, we investigate the continuum of one‐sign solutions of the nonlinear one‐dimensional Minkowski‐curvature equation with nonlinear boundary conditions by using unilateral global bifurcation techniques. We obtain the existence and multiplicity of one‐sign solutions for this problem and give the global structure of one‐sign solutions set according to different asymptotic behaviors of nonlinearity near zero.

Global bifurcation techniques for Yamabe type equations on Riemannian manifolds

We consider a closed Riemannian manifold (Mn,g) of dimension n≥3 and study positive solutions of the equation −Δgu+λu=λuq, with λ>0, q>1. If M supports a proper isoparametric function with focal varieties M1, M2 of dimension d1≥d2 we show that for any q<n−d2+2n−d2−2 the number of positive solutions of the equation −Δgu+λu=λuq tends to ∞ as λ→+∞. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. We also obtain multiplicity results for the Yamabe equation on Riemannian manifolds.

Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x&lt;R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) &gt; 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Positive solutions for some discrete semipositone problems via bifurcation theory

Let T > 1 be an integer, and let mathbb{T}={1, 2,ldots ,T}. We show the existence of positive solutions of the Dirichlet boundary value problem with second-order difference operator{−△2u(j−1)=λf(j,u(j)),j∈T,u(0)=u(T+1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\triangle ^{2} u(j-1)=\\lambda f(j, u(j)), \\quad j\\in \\mathbb{T},\\\\ u(0)=u(T + 1)=0, \\end{cases} $$\\end{document} where lambda >0 is a parameter, and f:mathbb{T}times mathbb{R} ^{+}to mathbb{R} is a continuous function satisfying f(j, 0)<0 for all jin mathbb{T}. The proofs of the main results are based upon topological degree and global bifurcation techniques.

Bifurcation results for a Kirchhoff type problem involving sign-changing weight functions

In this paper we establish some existence results for both compact and unbounded connected components of the positive solutions set of a nonlocal problem of the Kirchhoff type. The methods to show our main results are the local and global bifurcation techniques, a priori bounds for elliptic equation, and the properties of the principal eigenvalues. To obtain the a priori bounds for Kirchhoff type elliptic equation, some new techniques are introduced.

Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions

Abstract This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

Nonlinear elastic beam problems with the parameter near resonance

AbstractIn this paper, we consider the nonlinear fourth order boundary value problem of the form$$ \begin{array}\text \left\{ \begin{aligned} &amp;u^{(4)}(x)-\lambda u(x)=f(x, u(x))-h(x), \ \ x\in (0,1),\\ &amp;u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{aligned}\right. \end{array} $$which models a statically elastic beam with both end-points cantilevered or fixed. We show the existence of at least one or two solutions depending on the sign of λ−λ1, where λ1 is the first eigenvalue of the corresponding linear eigenvalue problem and λ is a parameter. The proof of the main result is based upon the method of lower and upper solutions and global bifurcation techniques.

Global Bifurcation from Intervals for the Monge-Ampère Equations and Its Applications

We shall establish the global bifurcation results from the trivial solutions axis or from infinity for the Monge-Ampère equations: det(D2u)=λm(x)-uN+m(x)f1(x,-u,-u′,λ)+f2(x,-u,-u′,λ), in B, u(x)=0, on ∂B, where D2u=(∂2u/∂xi∂xj) is the Hessian matrix of u, where B is the unit open ball of RN, m∈C(B¯,[0,+∞)) is a radially symmetric weighted function and m(r):=m(x)≢0 on any subinterval of [0,1], λ is a positive parameter, and the nonlinear term f1,f2∈C(B¯×R+3,R+), but f1 is not necessarily differentiable at the origin and infinity with respect to u, where R+=[0,+∞). Some applications are given to the Monge-Ampère equations and we use global bifurcation techniques to prove our main results.

On the Existence of Solutions for Impulsive Fractional Differential Equations

We are concerned with a type of impulsive fractional differential equations attached with integral boundary conditions and get the existence of at least one positive solution via global bifurcation techniques.

A priori bounds and existence of positive solutions for semilinear elliptic systems

We provide a-priori L∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions vp and uq for any (p,q) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system.

On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques

We are concerned with a type of fractional differential equations attached to boundary conditions. We investigate the existence of positive solutions and negative solutions via global bifurcation techniques.

Global structure of radial positive solutions for a prescribed mean curvature problem in a ball

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem div ( ϕ N ( ∇ v ) ) + λ f ( | x | , v ) = 0 in B ( R ) , v = 0 on ∂ B ( R ) , where ϕ N ( y ) = y 1 − | y | 2 , y ∈ R N , λ is a positive parameter, B ( R ) = { x ∈ R N : | x | < R } , and | ⋅ | denotes the Euclidean norm in R N . All results, depending on the behavior of nonlinear term f near 0, are obtained by using global bifurcation techniques.

Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications

We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems:x″″=αx++βx-+ratfx,0&lt;t&lt;1,x(0)=x(1)=x″(0)=x″(1)=0, wherer≠0is a parameter,a∈C([0,1],(0,∞)),x+=max⁡{x,0},x-=-min⁡{x,0},α,β∈C[0,1], andf∈C(R,R),sf(s)&gt;0,fors≠0. We give the intervals for the parameterrwhich ensure the existence of nodal solutions for the above fourth-order half-linear problems iff0∈[0,∞)orf∞∈[0,∞],wheref0=lims→0f(s)/sandf∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that(λk,0)is a bifurcation point of the above problems and there are two distinct unbounded continua,Ck+andCk-, consisting of the bifurcation branchCkfrom(μk,0), whereμkis thekth eigenvalue of the linear problem corresponding to the above problems. As the applications of the above result, we study the existence of nodal solutions for the following problems:x′′′′+kx′′+lx=rh(t)f(x), 0&lt;t&lt;1,x(0)=x(1)=x′(0)=x′(1)=0, wherer∈Ris a parameter andk,lare given constants;h(t)∈C([0,1],[0,∞))withh(t)≢0on any subinterval of[0,1]; andf:R→Ris continuous withsf(s)&gt;0fors≠0.We give the intervals for the parameterr≠0which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems iff0∈[0,∞]orf∞∈[0,∞],wheref0=lim|s|→0f(s)/sandf∞=lim|s|→+∞f(s)/s.We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem

We consider fourth-order boundary value problemsu′′′′(t)=λh(t)f(u(t)), 0&lt;t&lt;1, u(0)=∫01‍u(s)dα(s), u′(0)=u(1)=u′(1)=0, where∫01‍u(s)dα(s)is a Stieltjes integral withα(t)being nondecreasing andα(t)being not a constant on[0,1];h(t)may be singular att=0andt=1,h∈C((0,1),[0,∞))withh(t)≢0on any subinterval of(0,1);f∈C([0,∞),[0,∞))andf(s)&gt;0for alls&gt;0, andf0=∞, f∞=0, f0=lims→0+f(s)/s, f∞=lims→+∞f(s)/s.We investigate the global structure of positive solutions by using global bifurcation techniques.

Bifurcation from interval and positive solutions for a class of fourth-order two-point boundary value problem

We consider the fourth-order two-point boundary value problem x ���� + kx �� + lx = f (t, x), 0 < t <1 ,x(0) = x(1) = x � (0) = x � (1) = 0, which is not necessarily linearizable. We give conditions on the parameters k, l and f (t, x) that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques. MSC: 34B15

Bifurcation and positive solutions of a nonlinear fourth-order dynamic boundary value problem on time scales

This paper discusses the spectrum properties of a linear fourth-order dynamic boundary value problem on time scales and obtains the existence result of positive solutions to a nonlinear fourth-order dynamic boundary value problem. The key condition which makes nonlinear problem have at least one positive solution is related to the first eigenvalue of the associated linear problem. The proof of the main result is based upon the Krein-Rutman theorem and the global bifurcation techniques on time scales.

Positive periodic solutions of first-order delay differential equations with impulses

Using global bifurcation techniques, we show the existence and multiplicity of positive periodic solutions for a class of first-order delay differential equation with impulses.