Existence Of Positive Solutions Research Articles

Bifurcation analysis of fractional Kirchhoff problems with singular exponential nonlinearity

This paper deals with the bifurcation results of (weak) solutions for the Kirchhoff fractional p-Laplacian equation { M ( x , [ u ] s , p p ) ( − Δ ) p s u = f ( λ , x , u ) in Ω , u = 0 in R N ∖ Ω , where ( − Δ ) p s denotes the fractional operator, with sp = N, and the nonlinearity f exhibits the singular exponential growth at infinity. Moreover, the existence of unbounded components of (weak) solutions emanating from the trivial solution, is treated via the fix point result and the global bifurcation theorem due to Rabinowitz. Finally, the main feature of this paper consists of the existence of positive solutions to the above equation for λ small enough.

Positive solutions for a Hadamard-type fractional p-Laplacian integral boundary value problem

In this paper we study the existence of positive solutions to a Hadamard-type fractional integral boundary value problem using fixed point index. We construct a new linear operator and obtain our main results under some conditions concerning the spectral radius of this linear operator. Our method improves and generalizes some results in the literature.

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.

The existence of positive solutions for a critically coupled Schrödinger system in a ball of [formula omitted]

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u=v=0,x∈∂Ω, where Ω is a ball in R3, −λ1(Ω)<λ1,λ2<−14λ1(Ω), μ1,μ2>0 and β>0. Here λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary condition in Ω. We show that the problem has at least one positive solution for all 0<β<2μ1μ2. In particular, when λ1=λ2, we prove the existence of positive synchronized solutions for all β>0.

Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

On the existence of positive solutions of some nonlocal elliptic problems involving fractional p–Laplacian operator

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

On existence of positive solutions of p-Laplace equations

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δ p u = div(|∇u| p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \\ {0}, n ≥ 3, 1 < p 1, p 2 < n, q 1, q 2 > 0, and K 1(x), K 2(x) are double bounded functions.

Positive solutions of parameter-dependent nonlocal differential equations with convolution coefficients

In this paper we investigate a parameter-dependent nonlocal differential equations with convolution coefficients. Using the Birkhoff–Kellogg type theorem, existence of positive solutions is established. Under additional growth conditions, we obtain upper and lower bounds for the parameter. An example is also given to illustrate the main results.

Existence of positive solutions in the space of Hölder functions for a class of nonlinear fractional differential equations with integral boundary conditions

We study the existence of positive solutions for a fractional differential equation with integral boundary conditions. Our solutions are placed in the space of Hölder functions and the main tools used in the proof of the results are the classical Schauder fixed point theorem and a sufficient condition about the relative compactness in Hölder spaces. Moreover, some examples are shown illustrating the results.

Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions

Abstract We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.

On a class of infinite semipositone problems for ( p , q ) Laplace operator

We analyze a non-linear elliptic boundary value problem that involves ( p , q ) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in ( 0 , ∞ ) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving ( p , q ) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(ϑ,φ)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.

Normalized solutions for nonlinear Schrödinger equation involving potential and Sobolev critical exponent

In this paper, we consider the existence of positive solutions with prescribed L2-norm for the following nonlinear Schrödinger equation involving potential and Sobolev critical exponent{−Δu+V(x)u=λu+μ|u|p−2u+|u|4N−2u in RN,‖u‖2=a>0, where N≥3, μ>0, p∈[2+4N,2NN−2) and V∈C1(RN). Under different assumptions on V, we derive two different Pohozaev identities. Based on these two cases, we respectively obtain the existence of positive solution. As far as we are aware, we did not find any works on normalized solutions with Sobolev critical growth and potential V≢0. Our results extend some results of Wei and Wu (2022) [24] to the potential case.

Existence of Positive Solutions for a Singular Hessian Equation with a Negative Augmented Term

In this paper, we focus on the existence of positive solutions for a singular Hessian equation with a negative augmented term. By finding more appropriate upper and lower solutions, we not only overcome the difficulty due to the negative augmented term but also remove a critical condition required in the existing work and establish new results for the existence of positive solutions of the equations under study. Our results improve and complement many existing works.

Coexistence states in a cross-diffusion system of a competition model

The main goal of this paper is to study the stationary problem for a Lotka–Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by Kuto and Tsujikawa (2015) and Wang and Yan (2015), where the corresponding Neumann problem is analyzed.

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Existence of positive solutions of elliptic equations with Hardy term

This paper is devoted to studying the existence of positive solutions of the problem: ( ∗ ) { − Δ u = u p | x | a + h ( x , u , ∇ u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is an open bounded smooth domain with boundary ∂ Ω , and 1 &lt; p &lt; N − a N − 2 , 0 &lt; a &lt; 2 . Under suitable conditions of h ( x , u , ∇ u ) , we get a priori estimates for the positive solutions of problem ( ∗ ) . By making use of these estimates and topological degree theory, we further obtain some existence results for the positive solutions of problem ( ∗ ) when 1 &lt; p &lt; N − a N − 2 .

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Existence of solutions for asymptotically periodic quasilinear Schrödinger equations with local nonlinearities

This paper is concerned with the existence of positive solutions for asymptotically periodic quasilinear Schrödinger equations. By using a Nehari-type constraint and Moser iteration, we get the existence results which is a complement to the ones in Chu and Liu [Nonlinear Anal. Real World Appl. 44(2018), 118–127]. Moreover, we consider a new reformative asymptotic processes of the potential function and the nonlinearity term is only locally defined.