Exact Number Of Positive Solutions Research Articles

Dynamics of a predator–prey model with nonlinear growth rate and B–D functional response

In this paper, a predator–prey diffusive model, subject to homogeneous Dirichlet boundary conditions, with Beddington–DeAngelis functional response and nonlinear growth rate on the predator is proposed and the well posedness of its solution is systematically studied. Taking the capture rate m as the main parameter, we mainly investigate the existence, stability and exact number of positive solutions when m is large and other parameters meet a certain range of conditions. Meanwhile, some numerical simulations are applied to illustrate the analytical results. The main techniques used in this paper include the fixed point index theory, the super-sub solution method, the Lyapunov-Schmidt reduction procedure and the perturbation principle.

Exact Number of Positive Solutions for the Kirchhoff Equation

Exact Number of Positive Solutions for the Kirchhoff Equation

Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation \t\t\t{−(u′1−u′2)′=λf(u),x∈(−L,L),u(−L)=0=u(L),\\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}$$ \\textstyle\\begin{cases} - (\\frac{u'}{\\sqrt{1-u^{\\prime \\,2}}} )'=\\lambda f(u), &x\\in (-L,L), \\\\ u(-L)=0=u(L), \\end{cases} $$\\end{document} where λ and L are positive parameters, fin C[0,infty ) cap C^{2}(0,infty ), and f(u)>0 for 0< u< L. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f''(u)>0 and uf'(u)geq f(u)+frac{1}{2}u^{2}f''(u) for 0< u< L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

The number of positive solutions to the Brezis-Nirenberg problem

In this paper we are concerned with the well-known Brezis-Nirenberg problem { − Δ u = u N + 2 N − 2 + ε u , a m p ; in Ω , u &gt; 0 , a m p ; in Ω , u = 0 , a m p ; on ∂ Ω . \begin{equation*} \begin {cases} -\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, &amp;{\text {in}~\Omega },\\ u&gt;0, &amp;{\text {in}~\Omega },\\ u=0, &amp;{\text {on}~\partial \Omega }. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small ε &gt; 0 \varepsilon &gt;0 was obtained (see Monica Musso and Angela Pistoia [Indiana Univ. Math. J. 51 (2002), pp. 541–579]). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small ε &gt; 0 \varepsilon &gt;0 . By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green’s function of the domain Ω \Omega and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive ε \varepsilon , which depends also on Green’s function.

Positive solutions of a predator-prey model with cross-diffusion

In this paper, we consider the positive solutions for a predator-prey model with cross-diffusion and Holling type II functional response. In particular, the existence of positive solutions can be established by the bifurcation theory. Moreover, the uniqueness and the exact number of positive solutions is studied when the parameter m is large. Furthermore, for large cross-diffusion rate α with the spatial dimension is less than 5, we can derive the corresponding limit systems and also study the asymptotic behavior of positive solutions. Finally, some numerical simulations are presented to supplement the analysis results in one dimension case. This results give us some important information on the structure of positive solutions.

Synchronization of positive solutions for coupled Schrödinger equations

In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation $\left\{ {\begin{array}{*{20}{c}} {\Delta u - u + ({\mu _1}|u{|^p} + \beta |v{|^p})|u{|^{p - 2}}u = 0,}&amp;{{\text{i}}n\;{\mathbb{R}^n},} \\ {\Delta v - v + ({\mu _2}|v{|^p} + \beta |u{|^p})|v{|^{p - 2}}v = 0,}&amp;{{\text{i}}n\;{\mathbb{R}^n},} \end{array}} \right.$ where $ 2&lt; p&lt;\frac{n}{n-2}, $ if $ n\ge 3$ and $ 2&lt; p&lt;+∞ $, if $ n = 1, 2, $ and $μ_1, μ_2, β&gt;0 $ are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case $ n = 1 $ by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case $ p = 2 $ for which uniqueness of positive solutions was known ([19]).

Uniqueness of positive and compacton-type solutions for a resonant quasilinear problem

We study a one-dimensional $p$-Laplacian resonant problem with $p$-sublinear terms and depending on a positive parameter. By using quadrature methods we provide the exact number of positive solutions with respect to $\mu\in\mathopen{]}0,+\infty\mathclose[$. Specifically, we prove the existence of a critical value $\mu_1&gt; 0$ such that the problem under examination admits: no positive solutions and a continuum of nonnegative solutions compactly supported in $[0,1]$ for $\mu\in\mathopen{]}0,\mu_1\mathclose[$; a unique positive solution of compacton-type for $\mu=\mu_1$; a unique positive solution satisfying Hopf's boundary condition for $\mu\in\mathopen{]}\mu_1,+\infty\mathclose[$.

Time-map analysis to establish the exact number of positive solutions of one-dimensional prescribed mean curvature equations

We investigate the one-dimensional prescribed mean curvature equation with concave-convex nonlinearities in the form of − ( u ′ 1 + u ′ 2 ) ′ =λ( u p + u q ), u(x)>0, 0<x<1, u(0)=u(1)=0, where λ>0 is a parameter and p, q satisfy −1<p<q<+∞, and we obtain new exact results of positive solutions. Our methods are based on a detailed analysis of time maps.

Existence of Positive Solutions of One-Dimensional Prescribed Mean Curvature Equation

We consider the existence of positive solutions of one-dimensional prescribed mean curvature equation−(u′/1+u′2)′=λf(u),0&lt;t&lt;1,u(t)&gt;0,t∈(0,1),u(0)=u(1)=0whereλ&gt;0is a parameter, andf:[0,∞)→[0,∞)is continuous. Further, whenfsatisfiesmax{up,uq}≤f(u)≤up+uq,0&lt;p≤q&lt;+∞, we obtain the exact number of positive solutions. The main results are based upon quadrature method.

Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS

We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u<1,−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, and p,L>0 are two evolution parameters. We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.

Exact Number of Positive Solutions for a Class of Two-Point Boundary Value Problems

This paper considers the existence of positive solutions of the boundary value problems and , where and is a positive parameter. Using a time-map approach, we obtain the exact number of positive solutions in different cases.

Exact number of positive solutions for quasilinear boundary value problems with p-convex nonlinearity

By using the quadrature method, we study the exact number of positive solutions of the following quasilinear boundary value problem :
where 𝜑p (y) = |y|p-2 y, y ϵ R, p > 1, λ > 0 and ƒ : R+ ( R+ is of class C2 and p-convex function.

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.

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

We study the global bifurcation and exact multiplicity of positive solutions of{u″(x)+λfε(u)=0,−1<x<1,u(−1)=u(1)=0,fε(u)=−εu3+σu2−κu+ρ, where λ,ε>0 are two bifurcation parameters, and σ,ρ>0, 0<κ⩽σρ are constants. We prove the global bifurcation of bifurcation curves for varying ε>0. More precisely, there exists ε˜>0 such that, on the (λ,‖u‖∞)-plane, the bifurcation curve is S-shaped for 0<ε<ε˜ and is monotone increasing for ε⩾ε˜. Thus we are able to determine the exact number of positive solutions by the values of ε and λ. Our results extend those of Hung and Wang (K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., in press) from κ⩽0 to κ⩽σρ.

Exact Number of Positive Solutions for A Class of Two-Point Boundary Value Problem

Exact Number of Positive Solutions for A Class of Two-Point Boundary Value Problem

Exact number of solutions of stationary reaction–diffusion equations

We study both existence and the exact number of positive solutions of the problem ( P λ ) λ | u ′ | p - 2 u ′ ′ + f ( u ) = 0 in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ is a positive parameter, p > 1 , the nonlinearity f is positive in (0,1), and f ( 0 ) = f ( 1 ) = 0 . Assuming that f satisfies the condition lim s → 1 - f ( s ) ( 1 - s ) θ = ω > 0 where θ ∈ ( 0 , p - 1 ) , we study its behavior near zero, and we obtain existence and exactness results for positive solutions. We prove the results using the shooting method. We show that there always exist solutions with a flat core for λ sufficiently small. As an application, we prove the existence of a non-negative solution for a class of singular quasilinear elliptic problems in a bounded domain in R N having a flat core in a ball.

Exact number of solutions of a prescribed mean curvature equation

We study the exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equation − ( u ′ 1 + u ′ 2 ) ′ = λ ( u p + u q ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter and p, q satisfy either 1 < p < q < + ∞ or 0 < p < q < 1 .

Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

An exact multiplicity result of positive solutions for the boundary value problems , , , is achieved, where is a positive parameter. Here the function is and satisfies , for for some . Moreover, is asymptotically linear and can change sign only once. The weight function is and satisfies , for . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for lying in various intervals in . Moreover, we indicate how to extend the result to the general case.

On the exact number of solutions of a singular boundary-value problem

In this paper, we investigate the exact number of positive solutions of the Dirichlet boundary-value problem $u''-u^{-\gamma}+\beta=0$. We will show that the exact number of positive solutions can be 2, 1, 0, depending on the length of the interval and $\gamma$. This solves some open problems posed in [1].

On S-shaped bifurcation curves for a class of perturbed semilinear equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.