130 problems

Let $u$ solve … where $V$ is a real-valued bounded potential. Landis' conjecture. If, for some $C>0$ and any $\epsilon>0$, … then $u$ is trivial. The conjecture concerns quantitati…

Let $u$ solve … where $V\in W^{1,\infty}$, and let the rate of vanishing order refer to the maximal vanishing order of $u$. Kukavica's conjecture. The rate of vanishing order of…

Consider a bounded regular domain $\Omega\subset\mathbb{R}^n$ and the divergence-form equation … Here the coefficients $a^{ij}$ are bounded and measurable, and $u$ is a local weak…

Let $\Omega\subset\mathbb{R}^d$ be a Lipschitz domain, let $u$ be a bounded harmonic function in $\Omega$, and let $U\subset\partial\Omega$ be relatively open. Lin's conjecture. If…

Consider semilinear elliptic equations … with $u\in H^1(\mathbb{R}^n)$ and $u\not\equiv0$, under the natural structural conditions on $f$ considered by Berestycki and Lions. A radi…

Let two equal disks in $\mathbb{R}^2$ be separated by a distance parameter $\varepsilon$, and consider the Robin-type conductivity problem with perfect conductors and LC-type imper…

Logunov–Malinnikova's conjecture. Since the critical set where $| abla f|=0$ has codimension $2$, the correct condition on $E$ should be that it has positive $(n-2+delta)$-dimensio…

Let $n\geq 3$, $-\infty<a<\frac{n-2}{2}$, and $a\leq b<a+1$. Consider the critical Caffarelli–Kohn–Nirenberg equation … where $p=\frac{2n}{n-2+2(b-a)}$, and let … be the Felli–Schn…

Fernández-Real–Ros-Oton conjecture. Suppose that $u:\mathbb{R}^n\to(0,\infty)$ minimizes $I_1$ locally. If $n\leq n_e^*-1$, then $u$ is one-dimensional: in a suitable Euclidean coo…

Let $\Omega\subset\mathbb{R}^N$ be a domain and consider weak solutions of … Assume that the coefficient matrix satisfies … for $\xi\in\mathbb{R}^N$ and almost every $x\in\Omega$,…

Radial characterization conjecture. The solution $u$ is the minimal radial entire solution about some $x_*\in\mathbb{R}^N$ if and only if

Let $\mathcal{L}$ be an elliptic operator of order $2m$ on an open domain $\Omega\subset\mathbb{R}^N$, with $N\geq2$. For an open portion $\Gamma$ of $\partial\Omega$, impose homog…

Consider a positive function $u$ homogeneous of degree one in a domain $E gsubseteqmathbb{R}^n$, where $partial E$ is a cone, solving the one-phase free boundary problem … Assume t…

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a unimodular constraint manifold with negative chords that is asymptotic to a…

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a constraint manifold defined by an almost-complex structure tamed by $\Omega$…

Consider the singular elliptic problem described above, with potential coefficient $K$ and singular nonlinearity $g$ satisfying the stated hypotheses. The parameters $\beta$ govern…

Let $\boldsymbol\mu(z)=\mu(z)\,d\Sigma(z)$ be the associated $2$-form of a $C^\infty$-smooth real-valued function … boldsymbolmu(z)+frac{alpha}{2},mathbf K{mathbb H}(z)le 0,qquad z…

Let $\Omega\subset\mathbb{R}^N$ be the domain under consideration, let $p>1$, and let $\{u_n\}\subset C^1(\overline{\Omega})$ satisfy, in the weak sense, for some positive constant…

Classification conjecture. For $\beta=(d+\alpha)/(d-\alpha)-1$, the functions $u_{c,A}$ constitute all bounded solutions of the stationary equation, while for…

Codimension-two nodal-set conjecture. The nodal set of a solution of a generalized Dirac equation on $M$ has dimension

Positivity conjecture. The lower-order $Q$-curvature $Q_g^{(2k)}$ should be positive.

Let $u_\alpha$ be a ground state solution of problem, and let $V_\alpha$ be the associated potential. The maximum of $V_\alpha$ is attained at $1/2$. Concentration conjecture. Grou…

Let … and, for $a,c,d>0$, define … The constrained problem is to minimize $J_{a,c,d}(u)$ over $u\in E_2$ subject to $u(0,0)=1$. Ekeland–Nirenberg's positivity conjecture. For every…

Let $L = \sum_{i = 1}^{N} \ell_i p_i$ be a divisor with degree $\ell = \deg L$ even, and let $V$ denote the log-free variety associated with the polynomial system for the equation…

Let $\Omega$ be the domain in equations … , let $\partial\Omega$ denote its boundary, and suppose that $\mathfrak{B}:\mathbb{R}^{m+1}\to\mathbb{R}$ and…