Nonexistence Of Positive Solutions Research Articles

Positive solutions for systems of second-order integral boundary value problems

We investigate the existence and nonexistence of positive solutions of a system of secondorder nonlinear ordinary dierential equations, subject to integral boundary conditions. 2010 AMS Subject Classication: 34B10, 34B18.

On positive solutions of quasi-linear elliptic equations involving critical Sobolev growth

We study the boundary value problem of the quasi-linear elliptic equationdiv(|∇u|m−2∇u)+f(x,u,∇u)=0in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾2) is a connected smooth domain, and the exponent m∈(1,n) is a positive number. Under appropriate conditions on the function f, a variety of results on existence and non-existence of positive solutions have been established. This paper is a continuation of an earlier work Zou (2008) [18] of the author and, in particular, extends earlier results of Brezis and Nirenberg (1983) [3] for the semi-linear case of m=2, and of Pucci and Serrin (1986) [12] for the quasi-linear case of m≠2.

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ℝ + with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ≥ 3, ω > 0, 0 < T ≤ ∞, and u 0 ∈ C 0(ℝ + ). We prove the existence of a threshold number ω N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ℝ + .

On positive solutions for nonhomogeneous m-point boundary value problems with two parameters

This paper is concerned with the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the fixed-point theorem, the upper-lower solutions method, and the fixed-point index.

Positive solutions for singular boundary value problems involving integral conditions

We are interested in the following singular boundary value problem: { u ″ ( t ) + μ w ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where μ>0 is a parameter and ∫ 0 1 u(s)dA(s) is the Stieltjes integral. The function w∈C((0,1),(0,+∞)) and w may be singular at t=0 and/or t=1, f∈C([0,+∞),(0,+∞)) and f ∞ =+∞. Some a priori estimates and the existence, multiplicity and nonexistence of positive solutions are obtained. Our proofs are based on the method of global continuous theorem, the lower-upper solutions methods and fixed point index theory. Furthermore, we also discuss the interval of parameter μ such that the problem has a positive solution.

Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems

We consider the Hardy-Hénon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e., the nonexistence of positive solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a natural conjecture that this property should be true if and only if $(N+a)/(p+1)+(N+b)/(q+1)>(N-2)$. In this paper, we prove the conjecture for dimension $N=3$ in the case of bounded solutions and in dimensions $N\le 4$ when $a,b\le 0$, among other partial nonexistence results. As far as we know, this is the first optimal Liouville-type result for the Hardy-H\'enon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.

Positive solutions of boundary value problems for discrete and continuous beam equations

The authors study two related boundary value problems for discrete and continuous beam equations. Some upper and lower estimates for positive solutions of the problems are obtained. Sufficient conditions for the existence and nonexistence of positive solutions are established.

On a system of second-order multi-point boundary value problems

We study the existence of positive solutions for a system of nonlinear second-order ordinary differential equations subject to some multi-point boundary conditions. The nonexistence of positive solutions is also investigated.

Positive solutions for a class of higher order boundary value problems with fractional q-derivatives

The authors study the boundary value problem with fractional q-derivatives-(Dqνu)(t)=f(t,u),t∈(0,1),(Dqiu)(0)=0,i=0,…,n-2,(Dqu)(1)=∑j=1maj(Dqu)(tj)+λ,where q∈(0,1),m⩾1 and n⩾2 are integers, n-1<ν⩽n,λ⩾0 is a parameter, f:[0,1]×R→[0,∞) is continuous, ai⩾0 and ti∈(0,1) for i=1,…,m, and Dqν is the q-derivative of Riemann–Liouville type of order ν. The uniqueness, existence, and nonexistence of positive solutions are investigated in terms of different ranges of λ.

Positive Solutions of a Nonlinear Fourth‐Order Dynamic Eigenvalue Problem on Time Scales

Let 𝕋 be a time scale and a, b ∈ 𝕋, a &lt; ρ2(b). We study the nonlinear fourth‐order eigenvalue problem on 𝕋, , t ∈ [a, ρ2(b)] 𝕋, and obtain the existence and nonexistence of positive solutions when 0 &lt; λ ≤ λ* and λ &gt; λ*, respectively, for some λ*. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

Nonexistence of positive solutions of nonlinear boundary value problems

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity arguments, instead we use positivity properties of an associated linear operator which uses ideas related to the u0-positive operators of Krasnosel'skiuo.

Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions

In this paper, we study the existence and multiplicity of positive solutions of a class of nonlinear fractional boundary value problems with Dirichlet boundary conditions. By applying the fixed point theory on cones we establish a series of criteria for the existence of one, two, any arbitrary finite number, and an infinite number of positive solutions. A criterion for the nonexistence of positive solutions is also derived. Several examples are given for demonstration.

Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems

In this paper, we study a class of nonlinear boundary value problems in R + N . By means of minimax method and the Lusternik–Schnirelman category, the criteria of the existence, multiplicity and nonexistence of positive solutions are established.

Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters

By using the fixed-point index theory, we discuss the existence, multiplicity, and nonexistence of positive solutions for the coupled systems of Hammerstein integral equation with parameters.

Liouville-type theorems and bounds of solutions of Hardy–Hénon equations

We consider the Hardy–Hénon equation − Δ u = | x | a u p with p > 1 and a ∈ R and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space R N . It has been conjectured that this property is true if (and only if) p < p S ( a ) , where p S ( a ) is the Hardy–Sobolev exponent, given by ( N + 2 + 2 a ) / ( N − 2 ) . However, when N ⩾ 3 , the conjecture had up to now been proved only for a ⩽ 0 . Indeed the case a > 0 seems more difficult, due to p S ( a ) > ( N + 2 ) / ( N − 2 ) . In this paper, we prove the conjecture for a > 0 in dimension N = 3 , in the case of bounded solutions. Next, for the conjecture in the case a < 0 , and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem.

Positive Solutions of Multi-Point Boundary Value Problems for Second Order Impulsive Differential Equations with Deviating Arguments

In this article, we discuss the nonlinear multi-point boundary value problems for second order impulsive differential equations. Some sufficient conditions for the existence and nonexistence of positive solutions are proposed by using fixed point theorem in a cone.

Steady states for resource and sexual competition models with diffusions

We study the existence and nonexistence of positive solutions of resource and sexual competition models under a homogeneous Dirichlet boundary condition. In addition, when one of the coefficients of sexual competition is absent, the results for the multiplicity and stability of positive solutions are obtained. The methods employed are the upper-lower solution technique and the fixed point theory in a positive cone.

Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u0(.) ⩾ 0 and vanishing on the boundary ∂Ω × (0, T), given by where (resp. a Carnot Caratheodory metric ball in with smooth boundary and the time dependent singular potential function V(x, t) ∈ L1loc(Ω × (0, T)), , 1 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence

In this paper, we consider the nonlinear discrete boundary value problem { − Δ [ p ( t − 1 ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ Z [ 1 , T ] , u ( 0 ) = u ( T ) , p ( 0 ) Δ u ( 0 ) = p ( T ) Δ u ( T ) , where λ is a positive parameter. By using the fixed point index theory, the criteria of the existence, multiplicity and nonexistence of positive solutions are established in terms of different values of λ.

Existence, multiplicity and non-existence of positive solutions for two-point boundary-value problems with strong singularity

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problemswhere $\varphi_{p}(s)=|s|^{p-2}s$, $p&gt;1$, λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.