Existence Results Of Positive Solutions Research Articles

A Study on Positive Solutions to Nonlinear Fractional Differential Equations

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained.

Positive solutions for a Kirchhoff problem of Brezis–Nirenberg type in dimension four

We consider a Kirchhoff problem of Brezis–Nirenberg type in a smooth bounded domain of R4 with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters b and λ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.

Repeated S-shaped and Σ-shaped bifurcation curves for the one-dimensional non-autonomous Minkowski-curvature problem

We study the multiple existence results of positive solutions concerning one-dimensional non-autonomous Minkowski-curvature problems for certain ranges of a parameter. By providing a sufficient condition, we guarantee repeated S-shaped and repeated Σ-shaped global unbounded continua of positive solutions under superlinear and sublinear nonlinearities, respectively. This result is new even for autonomous cases.

Positive solutions for integral boundary value problems of nonlinear fractional differential equations

In this paper, we consider integral boundary value problems of nonlinear fractional differential equations. Existence results of positive solutions for the problem are obtained based on the Guo-Krasnoselskii theorem and the Five functional fixed point theorem. Simple examples follow the main results in successive sections.

Positive solutions for a system of fractional $ q $-difference equations with generalized $ p $-Laplacian operators

<abstract><p>In this paper, we consider the existence of positive solutions for a system of fractional $ q $-difference equations with generalized $ p $-Laplacian operators. By using Guo-Krasnosel'skii fixed point theorem, we obtain some existence results of positive solutions for this system with two parameters under some different combinations of superlinearity and sublinearity of the nonlinear terms. In the end, we give two examples to illustrate our main results.</p></abstract>

An Erdélyi-Kober fractional coupled system: Existence of positive solutions

<abstract><p>This paper studies an Erdélyi-Kober fractional coupled system where the variable is in an infinite interval, and the existence of positive solutions is considered. We first give proper conditions and then use the Guo-Krasnosel'skii fixed point theorem to discuss our problem in a special Banach space. The monotone iterative technique and the existence results of positive solutions for this system are established naturally. To show the plausibility of our main results, several concrete examples are given at the end.</p></abstract>

An Existence Result of Positive Solutions for the Bending Elastic Beam Equations

This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, 1] with the boundary condition u(0)=u(1)=u″(0)=u″(1)=0, which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity f(x, u, v) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(u, v)| is small or large enough. The discussion is based on the method of lower and upper solutions.

On the fractional Kirchhoff equation with critical Sobolev exponent

We provide four existence results of positive solutions to the fractional singular perturbation Kirchhoff equation using the finite-dimensional reduction approach and perturbed argument.

POSITIVE SOLUTIONS FOR A NONLOCAL PROBLEM WITH CRITICAL SOBOLEV EXPONENT IN HIGHER DIMENSIONS

This paper is devoted to a nonlocal problem involving critical Sobolev exponent and negative nonlocal term. By virtue of a cut-off technique and the concentration compactness principle, we prove the existence and asymptotic behavior of positive solutions for the considered problem. In particular, our results generalize the existence results of positive solutions to higher dimensions <i>N</i> ≥ 5.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function

Abstract In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

A priori bounds and existence of positive solutions to a p-Kirchhoff equations

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

Existence Results for a Nonlinear Fourth Order Ordinary Differential Equation with Four-Point Boundary Value Conditions

The aim of this paper is to study the more accurate existence results of positive solution for a nonlinear fourth order ordinary differential equation (for short NLFOODE) using four-point boundary value conditions (for short BVCs). The upper and lower solution method and Schauder’s fixed point theorem have been applied to demonstrate the obtained existence results. First, the Green’s function of the corresponding linear boundary value problem (for short BVP) has been constructed and then it is used to solve the considered BVP of this paper. An example has also been included at the end of this paper to support the analytic proof.

EIGENVALUE PROBLEM FOR FRACTIONAL DIFFERENCE EQUATION WITH NONLOCAL CONDITIONS

<abstract> In this work, we investigate a class of boundary value problem for fractional difference equation with nonlocal conditions <p class="disp_formula">$ \begin{cases}\Delta^{\nu}u(t)+\lambda f(t+\nu-1, u(t+\nu-1)) = 0, t\in\mathbb{N}_{0}^{b+1},\\ u(\nu-2) = h(u), \Delta u(\nu+b) = g(u), \end{cases} $ where $ 1&lt;\nu\leq2 $ is a real number, $ f:\mathbb{N}_{\nu-1}^{\nu+b}\times\mathbb{R}\rightarrow(0, +\infty) $ is a continuous function, $ g, h $ are given functionals, $ b\geq2 $ is an integer, $ \lambda&gt;0 $ is a parameter. By upper and lower solutions method, we can present the existence result of positive solutions. The eigenvalue intervals to this problem are studied by the properties of the Green function and Guo-Krasnosel'skii fixed point theorem in cones. </abstract>

Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

Existence results of positive solutions for Kirchhoff type biharmonic equationvia bifurcation methods

This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem $$ \left\{\begin{array}{ll} \Delta^{2}u-(a+b\int_\Omega \nabla u ^2dx)\triangle u=\lambda f(u(x)),\ \ \text{in}\ \Omega,\\ u=\triangle u=0,\ \ \text{on}\ \partial\Omega,\\ \end{array} \right. $$ where $\Omega\subset \mathbb{R}^{N}$($N\geq 1$) is a bounded domain with smooth boundary $\partial \Omega$, $a>0, b\geq 0$ are constants, $\lambda\in \mathbb{R}$ is a parameter. For the case $f(u)\equiv u$, we use an argument based on the linear eigenvalue problems of fourth order elliptic equations to show that there exists a unique positive solution for all $\lambda>\Lambda_{1,a}$, here $\Lambda_{1,a}$ is the first eigenvalue of the above problem with $b=0$; For the case $f$ is sublinear, we prove that there exists a positive solution for all $\lambda>0$ and no positive solution for $\lambda

Nonlinear flux “concave–convex” problems: a fibering method approach

This paper studies the nonlinear flux problem: where $$\varDelta _p$$ stands for the p-Laplacian operator, $$\varOmega \subset {\mathbb {R}}^N$$ is a bounded smooth domain, $$\lambda$$ is a positive parameter and $$\nu$$ stands for the outer unit normal at $$\partial \varOmega$$ . The exponents q, r are assumed to vary in the concave convex regime $$1< q< p < r$$ while $$1< p < N$$ and r is subcritical $$r < p^*$$ . Our objective here is showing the existence, for every $$0< \lambda < {{\bar{\lambda }}}$$ , of two different sets of infinitely many solutions of (P). The energy functional associated to the problem exhibits a different sign on each of these sets. The analysis of positive energy solutions involves the so-called fibering method (Drabek and Pohozaev in Proc R Soc Edinb Sect A 127(4):703–726, 1997). Our results have been inspired by similar ones in Garcia-Azorero et al. (J Differ Equ 198(1):91–128, 2004), Garcia-Azorero and Peral (Trans Am Math Soc 323(2):877–895, 1991) and El Hamidi (Commun Pure Appl Anal 3(2):253–265, 2004). This work can be considered as a natural continuation of Sabina de Lis (Differ Equ Appl 3(4):469–486, 2011), Sabina de Lis and Segura de Leon (Adv Nonlinear Stud 15(1):61–90, 2015) and Sabina de Lis and Segura de Leon (Nonlinear Anal 113:283–297, 2015). The main achievement of the latter of these works consisted in showing a global existence result of positive solutions to (P).

On supercritical problems involving the Laplace operator

AbstractWe discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

EXISTENCE OF POSITIVE SOLUTIONS TO A BOUNDARY VALUE PROBLEM FOR A DELAYED SINGULAR HIGH ORDER FRACTIONAL DIFFERENTIAL EQUATION WITH SIGN-CHANGING NONLINEARITY

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel'skii fixed point theorem, Leray-Schauder's nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.

POSITIVE SOLUTIONS FOR A &lt;i&gt;P&lt;/i&gt;-LAPLACIAN TYPE SYSTEM OF IMPULSIVE FRACTIONAL BOUNDARY VALUE PROBLEM&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ ^* $&lt;/tex-math&gt;&lt;/inline-formula&gt;

In this paper, the aim is to discuss a class of <i>p</i>-Laplacian type fractional Dirichlet's boundary value problem involving impulsive impacts. Based on the approaches of variational method and the properties of fractional derivatives on the reflexive Banach spaces, the existence results of positive solutions for our equations are established. Two examples are given at the end of each main result.

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii's Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.