On the Hénon-Lane-Emden conjecture
Mostafa Fazly Nassif Ghoussoub
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.
Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
Christine Chambers Nassif Ghoussoub
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
A variational principle for nonlinear transport equations
Nassif Ghoussoub
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
Remarks on multi-marginal symmetric Monge-Kantorovich problems
Nassif Ghoussoub Bernard Maurey
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
Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems
Nassif Ghoussoub
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.
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub
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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