Positive Stable Solutions Research Articles

Liouville‐type theorems for a nonlinear fractional Choquard equation

AbstractIn this paper, we are concerned with the fractional Choquard equation on the whole space with , and . We first prove that the equation does not possess any positive solution for . When , we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.

A nonexistence result of positive stable solutions to a weighted elliptic equation

We investigate the following weighted elliptic equation ??u + b(x) ? ?u = (1 + |x|2)?2 up in RN, where ? ? 0, p > 1, N ? 3 and the advection term b(x) is a smooth vector field satisfying certain decay condition. We establish a Liouville-type theorem for possible stable solutions of the equation above.

NONEXISTENCE OF STABLE SOLUTIONS OF THE WEIGHTED LANE-EMDEN SYSTEM

<abstract> The aim of this paper is to study the stability of the positive solutions of the weighted Lane-Emden system. By applying the structure of the $ m$-biharmonic weighted equation, we prove the nonexistence of positive stable solutions for the case $ 0&lt;p&lt;1&lt;p^{-1}&lt;\theta$. </abstract>

On positive stable solutions to weighted quasilinear problems with negative exponent

Let and , we prove the nonexistence of positive stable solutions of weighted quasilinear problem The result holds true for , or and , which is a positive critical exponent. Here and are nonnegative functions such that and for large |x|. We also construct an example to show the sharpness of our result.

Liouville type theorems for stable solutions of the weighted elliptic system

We examine the weighted elliptic system{−Δu=(1+|x|2)α2v,−Δv=(1+|x|2)α2up,inRN, and prove Liouville type theorems for the classical positive and nonnegative stable solutions in higher dimension. In particular, there are no positive stable solutions for any N≤12+5α, p>1 and α>0. Our proof is based on a combination of the bootstrap argument, Souplet's inequality and intermediate stability criterion, which is used to obtain sharp results.

Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(\xi)$ in the direction of any given unit vector $\xi$. A variational characterization of $c^*(\xi)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^*(\xi)$.

Stability of entire solutions to supercritical elliptic problems involving advection

We examine the equation given by (1)−Δu+a(x)⋅∇u=upin RN, where p>1 and a(x) is a smooth vector field satisfying some decay conditions. We show that for p<pc, the Joseph–Lundgren exponent, there is no positive stable solution of (1) provided one imposes a smallness condition on a along with a divergence free condition. In the other direction we show that for N≥4 and p>N−1N−3 there exists a positive solution of (1) provided a satisfies a smallness condition. For p>pc we show the existence of a positive stable solution of (1) provided a is divergence free and satisfies a smallness condition.

Liouville theorems for stable Lane–Emden systems and biharmonic problems

We examine the elliptic system given by1for 1 < p ⩽ θ and the fourth order scalar equation2where 1 < θ. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (1) (respectively, (2)) provided N ⩽ 10 and 2 ⩽ p ⩽ θ (respectively, N ⩽ 10 and 1 < θ). Results for higher dimensions are also obtained. These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions of3with u = v = 0 on . In particular there is no positive bounded solution of (3) for any 2 ⩽ p ⩽ θ if N ⩽ 11. Higher dimensional results are also obtained.

Multiple stable patterns for some reaction-diffusion equation in disrupted environments

We study the existence of multiple positive stable solutions for <p align="center"> $ -\epsilon^2\Delta u(x) = u(x)^2(b(x)-u(x)) \ \mbox{in}\ \Omega, \quad$ $ \frac{\partial u}{\partial n}(x) = 0 \ \mbox{on}\ \partial\Omega.$ <p align="left" class="times"> Here $\epsilon>0$ is a small parameter and $b(x)$ is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

A condition of the existence of stable positive steady-state solutions for a one predator two prey system

One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system. Under this condition we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem, discuss the positive stable solution problem bifurcated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.