Branch Of Positive Solutions Research Articles

Global bifurcation of positive solutions for a superlinear p -Laplacian system

We are concerned with the principal eigenvalue of (P) { − Δ p u = λ θ 1 φ p ( v ) , x ∈ Ω , − Δ p v = λ θ 2 φ p ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω and the global structure of positive solutions for the system (Q) { − Δ p u = λ f ( v ) , x ∈ Ω , − Δ p v = λ g ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω , where φ p ( s ) = | s | p − 2 s , Δ p s = div ( | ∇ s | p − 2 ∇ s ) , λ > 0 is a parameter, Ω ⊂ R N , N > 2 , is a bounded domain with smooth boundary ∂ Ω , f , g : R → ( 0 , ∞ ) are continuous functions with p -superlinear growth at infinity. We obtain the principal eigenvalue of ( P ) by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for ( Q ) via bifurcation technology.

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 .

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

A non‐local elliptic equation of Kirchhoff‐type for with Dirichlet boundary conditions is investigated for the cases where . It is well known that with the non‐local effect removed and , a branch of positive solutions bifurcates from infinity at and no positive solution exists whenever for some (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where is the principal eigenvalue of the linear problem . As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for . Moreover, regions with three positive solutions are found for small value of . Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.

Multiplicity for a strongly singular quasilinear problem via bifurcation theory

A [Formula: see text]-Laplacian elliptic problem in the presence of both strongly singular and [Formula: see text]-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive [Formula: see text]-direction and bifurcates from infinity at [Formula: see text]. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and in particular cases, global multiplicity.

Branches of positive solutions of a superlinear indefinite problem driven by the one-dimensional curvature operator

This paper aims at proving the existence and the localization of an unbounded connected set of positive regular solutions (λ,u) of the quasilinear Neumann problem −(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0,bifurcating from u=0 as λ→+∞. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, the weight a changes sign, and the function f is superlinear at 0. A novel approach is introduced based on the explicit construction of non-ordered sub and supersolutions.

On the uniqueness and monotonicity of solutions of free boundary problems

For any smooth and bounded domain Ω⊂RN, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on Ω in a neat interval depending only by the best constant of the Sobolev embedding H01(Ω)↪L2p(Ω), p∈[1,NN−2) and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for p>1, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for p=1. The threshold, which is sharp for p=1, yields a new condition which guarantees that there is no free boundary inside Ω. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.

The continuum branch of positive solutions for discrete simply supported beam equation with local linear growth condition

In this paper, we obtain the global structure of positive solutions for nonlinear discrete simply supported beam equation \t\t\tΔ4u(t−2)=λf(t,u(t)),t∈T,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=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} $$\\begin{aligned}& \\Delta ^{4}u(t-2)= \\lambda f\\bigl(t,u(t)\\bigr),\\quad t\\in \\mathbb{T}, \\\\& u(1)=u(T+1)=\\Delta ^{2}u(0)=\\Delta ^{2}u(T)=0, \\end{aligned}$$ \\end{document} with fin C(mathbb{T}times [0,infty ),[0,infty )) satisfying local linear growth condition and f(t,0)=0 uniformly for tin mathbb{T}, where mathbb{T}={2,ldots,T}, lambda >0 is a parameter. The main results are based on the global bifurcation theorem.

On branches of positive solutions for 𝑝-Laplacian problems at the extreme value of the Nehari manifold method

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p p -Laplacian, an indefinite nonlinearity, and depend on a real parameter λ \lambda . A special focus is given to the extreme value λ ∗ \lambda ^* of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter λ \lambda lies above the threshold λ ∗ \lambda ^* is obtained.

Existence of multiple solutions for a quasilinear elliptic problem

In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the $p$-derivative at zero and the $p$-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the $p$-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.

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.

Bifurcation in a multicomponent system of nonlinear Schrödinger equations

We consider the system $${-\Delta{u}_{j} + a(x)u_{j} = \mu_{j}u^{3}_{j} + \beta \sum_{k \neq j} u^{2}_{k}u_{j}}$$ , u j > 0, j = 1, . . . , n, on a possibly unbounded domain $${\Omega \subset \mathbb{R}^{N}, N \leq 3}$$ , with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose–Einstein condensates. We consider the self-focussing (attractive self-interaction) case $${\mu_{1}, \ldots, \mu_{n} > 0}$$ and take $${\beta \in \mathbb{R}}$$ as bifurcation parameter. There exists a branch of positive solutions with uj/uk being constant for all $${j, k \in \{1, \ldots, n\}}$$ . The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when n > 1 is odd).

Bifurcation of positive periodic solutions of first-order impulsive differential equations

We give a global description of the branches of positive solutions of first-order impulsive boundary value problem: which is not necessarily linearizable. Where is a parameter, are given impulsive points. Our approach is based on the Krein-Rutman theorem, topological degree, and global bifurcation techniques. MSC:34B10, 34B15, 34K15, 34K10, 34C25, 92D25.

Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem

Let \({T \in \mathbb{N}}\) be an integer with T ≥ 4. We give a global description of the branches of positive solutions of the nonlinear boundary value problem of fourth-order difference equation of the form $$\begin{array}{lll}\Delta^4 u(t-2)&=&f(t,u(t),\Delta^2u(t-1)),\quad t\in \{2,\ldots, T\},\\u(0)=&u(T+2)=\Delta^2u(0)=\Delta^2u(T)=0,\end{array}$$ that is not necessarily linearizable. Our approach is based on Krein–Rutman theorem, topological degree theory, and global bifurcation techniques.

Bifurcation from Interval and Positive Solutions of a Nonlinear Second‐Order Dynamic Boundary Value Problem on Time Scales

Let 𝕋 be a time scale with 0, T ∈ 𝕋. We give a global description of the branches of positive solutions to the nonlinear boundary value problem of second‐order dynamic equation on a time scale 𝕋, uΔΔ(t) + f(t, uσ(t)) = 0, t ∈ [0, T] 𝕋, u(0) = u(σ2(T)) = 0, which is not necessarily linearizable. Our approaches are based on topological degree theory and global bifurcation techniques.

Bifurcation interval for positive solutions to discrete second-order boundary value problems

Let be an integer with , , . We give a global description of the branches of positive solutions of the nonlinear eigenvalue problem which are not necessarily linearizable. Our approaches are based on topological degree and global bifurcation techniques.

Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations

This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Our nonlinearity may be singular at and/or .

Bifurcations of some elliptic problems with a singular nonlinearity via Morse index

We study the boundary value problem $\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1) where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem $$ - divA\left( {x,\nabla u} \right) = \lambda a\left( x \right)\left| u \right|^{p - 2} u + f\left( {x,u,\lambda } \right) $$ in Ω with p > 1. Under some proper assumptions on A(x, ξ), a(x) and f(x, u, λ), we show that the existence of an unbounded branch of positive solutions bifurcating from the principal eigenvalue of the problem −divA(x, ▿u) = λa(x)|u|p−2u.

Bifurcation and Stability of Travelling Waves in Self-focusing Planar Waveguides

Abstract A mathematical problem arising in the modelling of travelling waves in self- focusing waveguides is studied. It involves a nonlinear Schrödinger equation in ℝ1+1. In the paraxial approximation, the travelling waves are interpreted as standing waves of the NLS. Local and global bifurcation results are established for the corresponding \stationary" equation. The combination of variational arguments with an implicit function theorem yields smooth branches of positive solutions parametrized by the \frequency" of the standing wave of NLS. Precise informations on the asymptotic behaviour of various norms of the solutions along the branches are given. In the case of even symmetry, a global bifurcation result is proved, providing a branch of positive even solutions parametrized by frequencies in the whole half-line (0, ∞). The stability of the standing waves of the NLS relies on the monotonicity of their L2-norm as a function of the frequency. This criterion, precisely going back to formal arguments in the physical literature on self-focusing waveguides, is proved to hold along the whole global branch of solutions in the even case. The mathematical results are applied to the study of a non-homogeneous planar self-focusing waveguide with a Kerr-type nonlinear response, yielding existence of TE travelling waves and their stability among the set of TE modes. The bifurcation analysis gives informations on the power of the beam. In particular, it provides the possibility of low power cut-off.

Bifurcation from interval and positive solutions for second order periodic boundary value problems

We give a global description of the branches of positive solutions of second order periodic boundary value problems u ″ - q ( t ) u + λ f ( t , u ) = 0 , 0 < t < 2 π , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) , which are not necessarily linearizable. Our approach are based on topological degree and global bifurcation techniques.