14 problems

Let $v$ be a vector field satisfying the assumptions of the Liouville conjecture for stationary Navier–Stokes fields. In addition, suppose that $v$ is the restriction to…

Let $v$ be a vector field on $\mathbb R^{3}$, with pressure $p$, satisfying the stationary Navier–Stokes system … where $\nu>0$, together with … and…

Hénon–Lane–Emden conjecture. If $(p,q,a,b)$ is subcritical, then the system admits no positive solutions in $\left(C^2(\mathbb{R}^N\setminus\{0\})\cap C(\mathbb{R}^N)\right)^2$.

Lane–Emden conjecture. If $(p,q)$ is subcritical, then the system admits no positive solution in $\left(C^2(\mathbb{R}^N)\right)^2$.

Berestycki–Caffarelli–Nirenberg conjecture. (i) If there is a positive bounded solution $u$ of this problem, then $u$ depends only on $x_n$; consequently, $u_{x_n}>0$ and…

Consider the semilinear elliptic equation … where $f$ is a $C^1$ function, and let $u$ be a bounded stable $C^2$ solution, meaning that … for every…

Rotational symmetry conjecture. Every such solution $u$ on $\mathbb{R}^n$ is rotationally symmetric.

Nonexistence conjecture. If either of these integral conditions is satisfied on $M$, then every nonnegative solution of the equation referred to as $$ is identically zero.

Let $(M^n,g)$, $n\geq 3$, be a complete, locally conformally flat manifold with scalar curvature $R\geq 0$, and let $\Phi:M^n\to S^n$ be a conformal map. LCF Liouville conjecture.…

Regularity and minimal-growth solution conjecture. Then:

Let $u\in C^4(\mathbb{R}^n)$ be an entire 2-convex function, meaning that its Hessian eigenvalues lie in the closure of the $2$-Hessian ellipticity cone, and suppose that … through…

Liouville-type conjecture. Every weak solution $u\in\mathcal D^{1,p}_0({\mathbb R}_+^N)$ of this equation is identically zero:

Hé non-Lane-Emden conjecture. If $(u,v)$ is a nonnegative solution of the system and $(p,q)$ is under the critical hyperbola, then $u=v=0$. This is a Liouville-type conjecture for…

Let $X$) be the domain in Eq. (1), let $\hat X$ be its compactification, and let $\zeta\in\partial\hat X$. Assume that Eq. (1) has a Fuchsian type singularity at $\zeta$ and admits…