Global Bifurcation Of Positive Solutions Research Articles

Global bifurcation of positive solutions for a class of superlinear biharmonic equations

We are concerned with semilinear biharmonic equations of the form where B denotes the unit ball in ℝ N , N ≥ 1, λ > 0 is a parameter, ∂/∂ν is the outward normal derivative, f : ℝ → (0, +∞) is a continuous function that has super-linear growth at infinity. We use bifurcation theory, combined with an approximation scheme to establish the existence of an unbounded branch of positive radial solutions, which is bounded in positive λ–direction. If in addition, f satisfies certain subcritical condition, we show that the branch must bifurcate from infinity at λ = 0.

Global bifurcation of positive solutions of a quasilinear indefinite Neumann problem in some FLRW spacetimes

We are concerned with the global structure of the radial positive solution set of the radially symmetric prescribed curvature problem with the Neumann boundary condition in some Friedmann–Lemaître–Robertson–Walker spacetimes. The proof of our main result is based on bifurcation techniques.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We are concerned with the global bifurcation of positive solutions for semilinear elliptic systems of the form { − Δ u = λ f ( u , v ) in Ω , − Δ v = λ g ( u , v ) in Ω , u = v = 0 on ∂ Ω , where λ ∈ R is the bifurcation parameter, Ω ⊂ R N , N ≥ 2 is a bounded domain with smooth boundary ∂ Ω . We establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ -direction. The nonlinearities f , g ∈ C 1 ( R × R , ( 0 , ∞ ) ) are nondecreasing for each variable and have superlinear growth at infinity. The proof of our main result is based upon bifurcation theory. In addition, as an application for our main result, when f and g subject to the upper growth bound, by a technique of taking superior limit for components, then we may show that the branch must bifurcate from infinity at λ = 0 .

Global bifurcation of positive solutions for a class of superlinear first-order differential systems

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

Global bifurcation of positive solutions from zero in nonlinearizable elliptic problems with indefinite weight

This paper is devoted to the study of global bifurcation of solutions from zero of some nonlinearizable eigenvalue problems for second order elliptic partial differential equations with indefinite weight. The existence of global continua of solutions emanating from bifurcation intervals surrounding the trivial solutions corresponding to the principal eigenvalues of the linear problem obtained by setting the nonlinear term equal to zero and contained in the classes of positive and negative functions is shown.

Global Bifurcation of Positive Solutions of Semi-Linear Elliptic Partial Differential Equations with Indefinite Weight

In this paper we consider global bifurcation of solutions of nonlinear eigenvalue problems for second order uniformly elliptic equations with an indefinite weight function and the Dirichlet boundary condition. We show the existence of an unbounded continua contained in the classes of positive and negative functions and bifurcating from the points of the line of trivial solutions corresponding to the principal eigenvalues of the linearized problem.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We consider a system of semilinear equations of the form−Δu=λf(v)inΩ;−Δv=λg(u)inΩ;u=0=von∂Ω,} where λ∈R is the bifurcation parameter, Ω⊂RN; N≥2 is a bounded domain with smooth boundary ∂Ω. The nonlinearities f,g:R→(0,+∞) are nondecreasing continuous functions that have superlinear growth at infinity. We use bifurcation theory, combined with an approximation scheme, to establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ-direction.If in addition, Ω is convex and f,g∈C1 satisfy certain subcriticality condition, we show that the branch must bifurcate from infinity at λ=0.

Global Bifurcation of Positive Solutions of Asymptotically Linear Elliptic Problems

We are concerned with determining values ofλ, for which there exist positive solutions of the nonlinear elliptic problem-Δu=λa(x)f(u) in Ω,∂u/∂n+b(x)g(u)=0 on ∂Ω.The proof of our main results is based upon unilateral global bifurcation theorem of López-Gómez.

Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

This paper mainly dealt with the exact number and global bifurcation of positive solutions for a class of semilinear elliptic equations with asymptotically linear function on a unit ball. As byproducts, some existence and multiplicity results are also obtained on a general bounded domain.

Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications

We study the global bifurcation and exact multiplicity of positive solutions of { u '' (x) + lambda f(epsilon)(u) = 0, -1 f(epsilon) (u) = -epsilon u(3) + sigma u(2) + tau u + rho, where lambda, epsilon > 0 are two bifurcation parameters, and sigma, rho > 0, tau >= 0 are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying epsilon > 0. More precisely, we prove that, for any sigma, rho > 0, tau >= 0, there exists epsilon* > 0 such that, on the (lambda, parallel to u parallel to(infinity))- plane, the bifurcation curve is S-shaped for 0 = epsilon*. (We also prove the global bifurcation of bifurcation curves for varying lambda > 0.) Thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda. We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem { u '' (x) + lambda(-epsilon u(3) + u(2) + u + 1) = 0, -1 u(-1) = u(1) = 0, which was studied by Crandall and Rabinowitz (Arch. Rational Mech. Anal. 52 (1973), p. 177). In addition, we give an application to prove a conjecture of Smoller and Wasserman (J. Differential Equations 39 (1981), p. 283, lines 2-3) on the maximum number of positive solutions of a positone problem.

Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions

The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator–prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.

Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight

In this paper, we are concerned with the global structure and stability of positive solutions of periodic boundary value problem − u ″ ( t ) + q ( t ) u ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 2 π , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) , where q ∈ C ( R , [ 0 , ∞ ) ) is of periodic 2 π and q ( t ) ≢ 0 , t ∈ [ 0 , 2 π ] ; a ∈ C ( R , R ) is of periodic 2 π and changes sign. The proof of our main results are based on bifurcation techniques.

A food chain model for two resources in un-stirred chemostat

This paper deals with a multiple food chain model in un-stirred chemostat, where the prey feeds on two growth-limiting, nonreproducing resources. The conditions for existence of positive steady states are established. The global bifurcation of positive solutions is considered also. Furthermore, we obtain the conditions under which the predator possesses uniformly strong persistence or goes to extinction. The techniques used in this paper include the degree theory, the global bifurcation theory, the semigroup theory, and the maximum principle.

Bifurcation phenomena associated to a class of p -Laplacian like operators

Global bifurcation of positive solutions for some degenerate quasilinear elliptic problems is considered. The uniform estimate of the gradient of weak solutions is given. This estimate is crucial in our arguments.

Global Bifurcation of Positive Solutions in Some Systems of Elliptic Equations

In this paper the structure of the nonnegative steady-state solutions of a system of reaction-diffusion equations arising in ecology is investigated. The equations model a situation in which a predator species and a prey species inhabit the same region and the interaction terms are of Holling–Tanner type sothat the predator has finite appetite. Prey and predator birth-rates are treated as bifurcation parameters and the theorems of global bifurcation theory are adapted so that they apply easily to the system. Thus ranges of parameters are found for which there exist nontrivial steady-state solutions.