66 problems

Sirakov's equal-frequency uniqueness conjecture. In this equal-frequency setting, this is the unique positive solution.

Schaeffer's conjecture. Generically, the weak solution of the obstacle problem is also a strong solution, in the sense that the free boundary is a $C^\infty$ manifold. Equivalently…

Let $u$ solve the divergence-form elliptic equation … in $B(0,1)$, with … and … Let $Z(u)$ be the nodal set, $S(u)$ the singular set, and $\beta_u$ the frequency quantity used in t…

Let $v$ be a weight, let $Q$ be a non-negative definite symmetric matrix, and suppose that the Sobolev hypothesis holds for some $sigma>1$. Define … For data $f\in L^A(v;\Omega)$,…

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$. Let $u$ solve … and suppose there is a homeomorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ and a one-dimensional solution $U$ of the…

Let $\Omega$ and the parameters $a$, $b$, $c$, $\mu$, and $\lambda$ be as in Theorem 5.1, with $0\leqslant a<\frac{N-p}{p}$, and let the associated boundary-value problem be the eq…

The prescribed bounded mean curvature equation is considered for solutions defined on arbitrary manifolds. Heinz's continuity conjecture. Solutions to the prescribed bounded mean c…

Let $u$ be a smooth positive solution of equation (4.1), and let $K$ satisfy condition (4.2), with $a$ and $b$ as in that condition. Taliaferro's conjecture. If … then $u$ should h…

Gladiali–Grossi–Neves conjecture. Non-radial positive solutions may exist only when $\alpha=\alpha_k$ for some $k\in\mathbb N$.

Meyers' conjecture. Up to determining the optimal value of $c$, the optimal constant is conjectured to be $c=2$.

De Giorgi's conjecture. The lower bound $\exp(-c_nK)$ is highly suboptimal: the optimal Hölder exponent should have algebraic dependence on $K$, similarly to the examples of Piccin…

Comparison-principle conjecture. Under these assumptions, the comparison principle of the paper's system comparison theorem extends to solutions in $\mathcal{A}_g^k$. This conjectu…

Generalized convex-growth existence and uniqueness conjecture. Under these assumptions, the system admits a unique solution $u\in\mathcal{A}_g^k$. This conjecture proposes extendin…

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

Let $\Omega$ be a general bounded domain, and consider the Brezis–Nirenberg problem. A bubble-tower solution is a solution with multiple bubbles concentrating at the same point. Bu…

Wang–Wei classification conjecture. If $n\leq 7$, then $u$ is one-dimensional: either $u\equiv \pm1$, or

Let $(M^{n},g)$ be a closed manifold, let $u$ be a nonconstant eigenfunction satisfying … with $\lambda>0$, and let $Z(u)$ denote its nodal set. Yau's conjecture. There is a consta…

Let $\ell$ be the parameter appearing in the Grushin problem, and suppose the decay argument imposes the restrictive condition $\ell>4$. Technical-condition conjecture. The conditi…

The setting is that of parabolic operators of type $\mathcal{L}$ in the upper half-space, with coefficients satisfying a weak, also called $L^2$, Dahlberg–Kenig–Pipher condition;…

Let $a_{ij}(\cdot)$ be the coefficient matrix in the framework above, let $\overline{a}_{ij}$ denote its homogenized coefficients, and let $\beta_0$ be the critical value. Write…

Let $f$ be the nonlinearity in the radial equation … that has precisely $k-1$ nodes. This conjecture concerns the nodal characterization of bound states and asserts uniqueness with…

Let $\Omega$ be the domain, let $f\in L^1(\Omega)$ be nonnegative, and let $u$ be a solution of the singular elliptic problem … with homogeneous Dirichlet boundary condition $u=0$…

Let $\mathbf{w}^0$ be a non-degenerate phase-separating solution of the problem under consideration in a domain $\Omega\subset\mathbb{R}^N$, where $N\geq 3$. Non-degeneracy means t…

Let $\Omega\subset\mathbb R^3$ be a bounded set, let $q\in(2,4)$, and consider the Dirichlet problem … with parameter $\lambda>0$. Brezis–Nirenberg conjecture. There exists a const…

Non-degeneracy conjecture. The non-degeneracy property holds without the assumption that $\gamma$ is sufficiently small.