DCDS
On the Hénon-Lane-Emden conjecture
Mostafa Fazly Nassif Ghoussoub
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2513-2533 doi: 10.3934/dcds.2014.34.2513
We consider Liouville-type theorems for the following Hénon-Lane-Emden system \begin{eqnarray*} \left\{ \begin{array}{lcl} -\Delta u&=& |x|^{a}v^p \ \ in\ \ \mathbb{R}^n,\\ -\Delta v&=& |x|^{b}u^q \ \ in\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. 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 be bounded, extending a result established recently by Phan-Souplet in the scalar case.
    Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $n\ge 3$ in the first case (resp., $n\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1< p < \frac{n+2+2a}{n-2}$ (resp., $ 1< p < \frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \begin{equation*}\label{sysdim00} n < 2 + 2 (\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}} + \sqrt{ \frac{pq(q+1)}{p+1} - \sqrt{\frac{pq(q+1)}{p+1}}}). \end{equation*}
keywords: stable solutions Liouville theorems nonlinear elliptic systems finite Morse index solutions Hénon-Lane-Emden conjecture.
DCDS
Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
Christine Chambers Nassif Ghoussoub
Discrete & Continuous Dynamical Systems - A 2002, 8(1): 267-281 doi: 10.3934/dcds.2002.8.267
A general theorem on the multiplicity of critical points for non-invariant deformations of symmetric functionals is established, using a method introduced by Bolle [5]. This result is used to find conditions sufficient for the existence of multiple solutions of semi-linear elliptic partial differential equations of the form

$-\Delta u = p(x, u) + f(\theta, x, u)\quad $ on $\Omega$

$u = 0\quad$ on $\partial \Omega$

where $p(x, \cdot)$ is odd and $f$ is a perturbative term. An application of this result is the problem

$-\Delta u = \lambda |u|^{q-1}u + |u|^{p-1}u + f\quad$ on $\Omega$

$u = u_0\quad$ on $\partial \Omega$

where $\Omega$ is a smooth, bounded, open subset of $\mathbf R^n (n \geq 3), \lambda > 0, 1\leq q < p, f \in C(\bar \Omega, \mathbf R)$ and $u_0\in C^2(\partial \Omega, \mathbf R)$. It is proven that this equation has an infinite number of solutions for $p < \frac{n+1}{n-1}$ and that for any sub-critical $p$ i.e., $p < \frac{n+2}{n-2}$, there are as many solutions as we like, provided $||f||_{frac{p+1}{p}}$ and $||u_0||_{p+1}$ are small enough.

keywords: Critical point theory perturbation from symmetry min-max method non-linear multiplicity elliptic PDEs. variational methods
CPAA
A variational principle for nonlinear transport equations
Nassif Ghoussoub
Communications on Pure & Applied Analysis 2005, 4(4): 735-742 doi: 10.3934/cpaa.2005.4.735
We verify -after appropriate modifications- an old conjecture of Brezis-Ekeland [4] concerning the feasibility of a global and variational approach to the problems of existence and uniqueness of solutions of non-linear transport equations, which do not normally fit in an Euler-Lagrange framework. Our method is based on a concept of "anti-self duality" that seems to be inherent in many problems, including gradient flows of convex energy functionals treated in [10] and other parabolic evolution equations ([7]).
keywords: Transport equation convex duality
DCDS
Remarks on multi-marginal symmetric Monge-Kantorovich problems
Nassif Ghoussoub Bernard Maurey
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1465-1480 doi: 10.3934/dcds.2014.34.1465
Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
keywords: $m$-cyclically antisymmetric functions. Monge-Kantorovich duality $m$-cyclically monotone vector fields Mass transport
DCDS
Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems
Nassif Ghoussoub
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 187-220 doi: 10.3934/dcds.2008.21.187
Selfdual variational theory -- developed in [8] and [9] -- allows for the superposition of appropriate "boundary" Lagrangians with "interior" Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in many natural settings, leading to a variational resolution of certain differential systems. These results are applied to nonlinear transport equations with prescribed exit values, Lagrangian intersections of convex-concave Hamiltonian systems, initial-value problems of dissipative systems, as well as evolution equations with periodic and anti-periodic solutions.
keywords: Selfdual functionals boundary value and initial-value problems.
DCDS
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1033-1050 doi: 10.3934/dcds.2010.28.1033
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \R^N$, with the Navier boundary condition $ u=\Delta u =0 $ on δΩ. We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution u * is smooth provided $N\leq 5$. If in addition $\lim$i$nf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then u * is regular for $N\leq 7$, while if $\gamma$:$= \lim$s$up_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$. It follows that u * is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and $N< \frac{8p}{p-1}$. We also show that if $ f(t) = (1-t)^{-p}$, $p>1$ and $p\ne 3$, then u * is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., u * is smooth for $ N \le 12$ when $ f(t) = e^t$ [11], and for $ N \le 8$ when $ f(t) = (1-t)^{-2}$ [9] (see also [22]).
keywords: Stable solution Superlinear and Singular nonlinearities. Extremal solution

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