Multiple Positive Solutions Research Articles

Catalogue of existence of the multiple physical solutions of hydromagnetic flow over a stretching/shrinking sheet for viscoelastic second-grade and Walter's B fluids

In this work, the author restudied a fourth-order nonlinear ordinary differential equation that commonly appears in investigating the hydromagnetic flow of two types of viscoelastic fluids over a stretching/shrinking sheet. In this type of problem, vital question arises: what are the parameters values needed to obtain regions and numbers of the physical solutions? To answer this question, the researchers worked on suggesting/estimating the initial values, or even chose them randomly. This usually leads to wasting a lot of time, effort and maybe to untrustworthy results. Towards avoiding this rugged road, the present work answered this question decisively and definitively. In particular, an explicit and direct catalogue of existence multiple real positive solutions was in detail introduced. MATHEMATICA code was programmed and, therefore, many tables and figures of the critical values and regions were produced at different values of the controlled parameters. In addition, and maybe for the first time, the mathematical expressions of these critical values were secured. Hence, remarkable differences were detected between the existing results and those obtained in the literature, therefore, spurious physical sights were gained. In conclusion, this research determined the whole values of the parameters that are required to control the number of solutions or even not to obtain solutions at all. Therefore, the current catalogue aims to help the researchers to avoid guessing the parameters’ values on solving the current type of problem.

Multiple positive solutions for a $(p,q)$-Laplace equation involving singular and critical terms

In this paper, we study a $(p,q)$-Laplace equation with singular and critical terms and establish the existence of multiple positive solutions via use of variational methods.

Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

It is well known that a single nonlinear fractional Schrödinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) when the positive parameter ε is sufficiently small [see G. Chen and Y. Zheng, Commun. Pure Appl. Anal. 13, 2359 (2014); J. Dávila et al., J. Differential Equations 256, 858 (2014); and M. M. Fall et al., Nonlinearity 28, 1937 (2015)]. In this work, we find multiple different positive solutions for a class of weakly coupled fractional Schrödinger systems with two potentials V1(x) and V2(x) that have some of the same minimum points; the positive solutions are concentrated at these minimum points. By using energy estimates, the Nehari manifold technique, and the Lusternik–Schnirelmann theory of critical points, we obtain some multiplicity results for a class of weakly coupled fractional Schrödinger systems. The existence and nonexistence of least energy positive solutions are also explored.

Positive solutions for a class of two-term fractional differential equations with multipoint boundary value conditions

In this article, we study the existence of positive solutions to a class of two-term fractional nonlocal boundary value problems. The existence and multiplicity of positive solutions are established by means of fixed point index theory. The nonlinearity f(t,x) permits a singularity at t = 0,1 and x=0.

Existence of Positive Solutions to Singular Boundary Value Problems Involving φ-Laplacian

This paper is concerned with the existence of positive solutions to singular Dirichlet boundary value problems involving φ -Laplacian. For non-negative nonlinearity f = f ( t , s ) satisfying f ( t , 0 ) ≢ 0 , the existence of an unbounded solution component is shown. By investigating the shape of the component depending on the behavior of f at ∞ , the existence, nonexistence and multiplicity of positive solutions are studied.

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.

Multiple positive solutions for the critical Kirchhoff type problems involving sign-changing weight functions

This paper is concerned with the multiplicity of positive solutions for the critical Kirchhoff type problems involving indefinite weight functions(0.1){−M(∫Ω|∇u|2dx)Δu=Qλ(x)|u|q−2u+K(x)|u|2⁎−2u,x∈Ω,u=0,x∈∂Ω, where Ω is a smooth bounded domain in RN(N≥3), 1<q<2, M(s)=a+bsβ with β>0,a>0 and b>0, the weight functions Qλ and K are continuous and changing-sign. Using the Nehari manifold, fibering maps and Ljusternik-Schnirelmann category, we prove that at least two positive solutions for (0.1) exist provided that β=1 and 2⁎≥4. Furthermore, by the mountain pass theorem and Ekeland's variational principle, it is shown that (0.1) possesses at least three positive solutions whenever β>2N−2, including the case that β=1 and 2⁎<4. Our results generalize some recent results in the literature.

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Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

Existence and multiplicity of positive solutions for a class of singular fractional nonlocal boundary value problems

In this article, we consider a class of singular fractional differential equations with nonlocal boundary value conditions. The existence and multiplicity of positive solutions are derived by the fixed point index theory, and the nonlinearity f(t,x) may be singular at t=0,1 and x=0. The interesting point is that the existence results are closely associated with the relationship between 1 and the spectral radii corresponding to the relevant linear operators. An example is also given to demonstrate the validity of the main results.

A four-point boundary value problem with singular $$\phi $$-Laplacian

We prove that the four-point boundary value problem $$\begin{aligned} -\left[ \phi (u') \right] ^{\prime }=f(t,u, u'), \quad u(0)=\alpha u(\xi ), \quad u(T)=\beta u(\eta ), \end{aligned}$$ where $$f:[0,T] \times \mathbb {R}^2 \rightarrow \mathbb {R}$$ is continuous, $$\alpha , \; \beta \in [0,1)$$ , $$0<\xi< \eta <T $$ , and $$\phi :(-a,a) \rightarrow \mathbb {R}$$ ( $$0<a<\infty $$ ) is an increasing homeomorphism, which is always solvable. When instead of f is some $$g:[0,T] \times [0, \infty ) \rightarrow [0, \infty )$$ , we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.

Multiple Positive Solutions for Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions and a Parameter

By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.

Vector solutions for two coupled Schrödinger equations on Riemannian manifolds

We are interested in two coupled nonlinear Schrödinger equations on a smooth compact boundaryless Riemannian manifold, which arises from Bose-Einstein condensates and nonlinear optics theory. The existence of least energy solutions, multiplicity of positive solutions, as well as synchronized vector solutions is obtained.

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem involving two mean curvature equations of spacelike graphs in Euclidean and Minkowski spaces.

Multiple positive solutions to Kirchhoff equations with competing potential functions in mathbb{R}^{3}

In this paper, we study the existence of multiple positive solutions to the following Kirchhoff equation with competing potential functions: \t\t\t{−(ε2a+εb∫R3|∇v|2)Δv+V(x)v=K(x)|v|p−1vin R3,v>0,v∈H1(R3),\\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}$$ \\textstyle\\begin{cases} -(\\varepsilon ^{2}a+\\varepsilon b{\\int _{\\mathbb{R}^{3}}}{ \\vert \\nabla v \\vert } ^{2})\\Delta v+V(x)v=K(x) \\vert v \\vert ^{p-1}v \\quad \\mbox{in }\\mathbb{R}^{3}, \\\\ v>0, \\quad v\\in H^{1}(\\mathbb{R}^{3}), \\end{cases} $$\\end{document} where varepsilon >0 is a small parameter, a,b>0 are constants, 3< p<5. We relate the number of solutions with the topology of the global minima set of the function {V^{frac{2}{p-1}-frac{1}{2}}(x)}/ {K^{frac{2}{p-1}}(x)}. The Nehari manifold and Ljusternik–Schnirelmann category theory are applied in our study.

Existence of multiple positive solutions for fractional Laplace problems with critical growth and sign-changing weight in non-contractible domains

We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.

Multiple positive solutions for perturbed nonlinear fractional differential system with two control parameters

In this paper, we study a class of perturbed nonlinear fractional differential systems with two control parameters. Under some appropriate conditions on the primitive function of nonlinear terms F_{u} and F_{v}, we prove the multiplicity of weak solutions by using the variational method and Ricceri’s critical points theorems.

Positive periodic solutions of functional discrete systems with a parameter

The existence of multiple positive periodic solutions of the system of difference equations with a parameter x(n + 1) = A(n, x(n))x(n) + λf(n, xn), is studied. In particular, we use the eigenvalue problems of completely continuous operators to obtain our results. We apply our results to a well-known model in population dynamics.

Multiple positive solutions for a coupled system of $p$-Laplacian fractional order three-point boundary value problems

In this paper, we attempt to establish the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order boundary value problems \begin{aligned}\begin{cases} D_{a^+}^{\beta _1}(\phi _{p}(D_{a^+}^{\alpha _1}u(t)))=f_1(t, u(t), v(t)) & \quad a\lt t\lt b,\\ D_{a^+}^{\beta _2}(\phi _{p}(D_{a^+}^{\alpha _2}v(t)))=f_2(t, u(t), v(t)) & \quad a\lt t\lt b, \end{cases}\end{aligned} with the boundary conditions \begin{aligned}\begin{cases} u^{(j)}(a)=0,\ j=0,1,2,\ldots , \ u{''}(b)=\delta u{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _1}u(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _1} u(b)) =\vartheta \phi _{p}( D_{a^+}^{\alpha _1}u(\eta )),\\ v^{(j)}(a)=0,\ j=0,1,2,\ldots , \ v{''}(b)=\delta v{''}(\xi ),\\ \phi _{p}(D_{a^+}^{\alpha _2}v(a))=0, \ \phi _{p}(D_{a^+}^{\alpha _2} v(b))=\vartheta \phi _{p}( D_{a^+}^{\alpha _2}v(\eta )), \end{cases}\end{aligned} by applying the five functional fixed point theorem.

Multiple positive solutions for nonlinear mixed fractional differential equation with p-Laplacian operator

In this article, multiple positive solutions are considered for nonlinear mixed fractional differential equations with a p-Laplacian operator. Using the Avery–Peterson fixed point theorem, we conclude to the existence of positive solutions for the fractional boundary value problem. An example is also presented to illustrate the effectiveness of the main result.