46 problems

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

Consider the Lane–Emden problem … where $\mathbb R^n_+=\{x_n>0\}$, $n\geq 2$, and $q>1$. Liouville conjecture. The only nonnegative solution is the trivial one, namely $u\equiv 0$.…

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

Koch–Nadirashvili–Seregin–Šverák conjecture. Every bounded mild ancient solution of the axi-symmetric Navier–Stokes equations is constant.

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

Consider the stationary Navier–Stokes system in $\mathbb{R}^3$: … Here $u$ is the velocity vector field, $\pi$ is the pressure, and $\dot{H}^1(\mathbb{R}^3)$ denotes the homogeneou…

GHS23 divergence conjecture. If

Grigor'yan–Sun–Verbitsky conjecture. If

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…

Yau's conjecture. The space $\mathcal{H}_d(M)$ should have finite dimension.

Bidaut-Véron–Huidobro–Véron conjecture. Any solution $u$ must be constant in the second critical range.

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…

Liouville nonexistence conjecture. The nonexistence result for (1.3) should hold whenever

Rigidity conjecture. If $u$ is positive, then $(M,g)$ is isometric to Euclidean space and $u$ is an Aubin–Talenti bubble, namely

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…

Let $\mathscr{C}_{\infty}$ be the infinite cluster of supercritical bond percolation on $\operatorname{\mathbb{Z}}^d$, and let $u:\mathscr{C}_{\infty}\to\operatorname{\mathbb{R}}$…

Let $L$ be an arbitrary Lévy operator satisfying the assumption of Theorem, and let $\check{H}_D$ denote the corresponding harmonic measure for a bounded open set $D$. A support is…

Let $p,q$ satisfy $pq>1$, and consider system (1) in $\mega=\mathbb{R}^{N}$, without imposing a sign condition on the solutions. Liouville conjecture. Every solution of this system…

Uniqueness conjecture. The equation has only one nonnegative solution, namely $u=0$. This is a Liouville-type assertion for the Lane–Emden equation in the half-space; the supplied…

The bounded ancient-solution Liouville conjecture. The solution $f$ must be a stationary Maxwellian. This is intended to rule out non-Maxwellian bounded ancient solutions arising f…

Let $f\colon \mathbb{R}^n\to\mathbb{R}^n$ be a solution of the higher-dimensional inequality … where $\Sigma$ is locally integrable. The inequality is a higher-dimensional analogue…

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: