On the H\'enon-Lane-Emden conjecture

@article{Fazly2011OnTH,
  title={On the H\'enon-Lane-Emden conjecture},
  author={Mostafa Fazly and Nassif Ghoussoub},
  journal={arXiv: Analysis of PDEs},
  year={2011}
}
We consider Liouville-type theorems for the following Henon-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to… 
Liouville theorems for the polyharmonic Henon-Lane-Emden system
We study Liouville theorems for the following polyharmonic H\'{e}non-Lane-Emden system \begin{eqnarray*} \left\{\begin{array}{lcl} (-\Delta)^m u&=& |x|^{a}v^p \ \ \text{in}\ \ \mathbb{R}^n,\\
Liouville type theorem for critical order Lane-Emden-Hardy equations in $\mathbb{R}^n$.
In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-\Delta)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\,
Liouville type theorems, a priori estimates and existence of solutions for sub-critical Order Lane—Emden—Hardy equations
We study the sub-critical order Lane—Emden—Hardy equations (0.1) $${\left( { - {\rm{\Delta }}} \right)^m}u\left( x \right) = {{{u^p}\left( x \right)} \over {{{\left| x
Monotonicity formula and Liouville-type theorems of stable solution for the weighted elliptic system
In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad
A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type
We prove there are no positive solutions with slow decay rates to higher order elliptic system \begin{eqnarray} \left\{ \begin{array}{c} \left( -\Delta \right) ^{m}u=\left\vert x\right\vert
Liouville-Type Theorems for Higher Order Elliptic Systems
We prove there are no positive solutions to higher order elliptic system \begin{equation*} \left\{ \begin{array}{c} \left( -\Delta \right) ^{m}u=v^{p} \\ \left( -\Delta \right) ^{m}v=u^{q}
Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^{n}$
In this paper, we consider the critical order Hardy-Henon equations \begin{equation*} (-\Delta)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb{R}^{n},
A Liouville-Type Theorem for Fractional Elliptic Equation with Exponential Nonlinearity
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$\begin{aligned} (-\Delta )^s u=e^u \text{ in } {\mathbb {R}}^{N}, \end{aligned}$$ ( - Δ ) s u = e u in R N
...
...

References

SHOWING 1-10 OF 55 REFERENCES
De Giorgi type results for elliptic systems
AbstractWe consider the following elliptic system $${\Delta}u = \nabla H (u) \quad {\rm in}\quad \mathbf{R}^N,$$where $${u : \mathbf{R}^N \to \mathbf{R}^m}$$ and $${H \in C^2(\mathbf{R}^m)}$$, and
Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations
By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation \[ -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq
Liouville theorems for stable solutions of biharmonic problem
We prove some Liouville type results for stable solutions to the biharmonic problem $$\Delta ^2 u= u^q, \,u>0$$ in $$\mathbb{R }^n$$ where $$1 < q < \infty $$. For example, for $$n \ge 5$$, we show
A new dynamical approach of Emden-Fowler equations and systems
We give a new approach on general systems of the form \[ (G)\left\{ \begin{array} [c]{c}% -\Delta_{p}u=\operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon_{1}\left\vert
Liouville-type theorems and bounds of solutions for Hardy-Hénon elliptic systems
Abstract. We consider the Hardy-Hénon system −∆u = |x|v, −∆v = |x|u with p, q > 0 and a, b ∈ R and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive
The proof of the Lane–Emden conjecture in four space dimensions
Liouville Type Theorems for Stable Solutions of Certain Elliptic Systems
Abstract We establish Liouville type theorems for elliptic systems with various classes of nonlinearities on ℝN. We show, among other things, that a system has no semi-stable solution in any
NONEXISTENCE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC SYSTEMS IN R N
has been studied by a number of authors; see for example [2, 4, 5, 9, 10, 13, 14, 16] and the references therein. In particular in Gidas-Spruck ([5]) it is proved among other things that if f(x, u) =
...
...