25 problems

Let $E$ be a Banach space whose current point is … and let $B\subseteq E$ be closed. The following should hold simultaneously: the elements of $B$ satisfy at $t=0$ the initial cond…

Let $s_c$ be the scaling-critical Sobolev index of a partial differential equation, meaning that the homogeneous $H^{sc}$-norm is invariant under the equation's scaling symmetry.…

Consider the pseudoconformal blow-up solutions of the self-dual Chern–Simons–Schrödinger equation and the rotational instability described by the phase rotation and subsequent spre…

Let $K$ be a finite subset of $cmathbb{R}^{n\times m}$, and let $\operatorname{conv}_\mathrm{rc}K$ denote its rank-one convex hull, namely the set of matrices satisfying…

Small-mass negativity conjecture. In this range, there exists $c_0=c(N,p,q,b)>0$ such that

Let $N$, $q$, $b$, and $p$ be parameters as in the paper, and let obreak 2q^ and $(N-q)^+$ denote the corresponding critical exponents and positive part. For $c0$, let…

MA-positivity conjecture. Any MA-semipositively curved solution of the vector bundle Monge–Ampère equation is MA-positively curved.

Let $D$ be a polydisc, let $p\in\mathbb{N}^{\times}$, and let $\upsilon_j(\boldsymbol{x})$ for $0\leqslant j\leqslant p$ be holomorphic on $D\cap(\{0\}\times\mathbb{C}^n)$, with…

Consider the ramified initial-value problem … … where $c_1,c_2\in\mathbb{C}^{\times}$, $c_3\in\mathbb{C}$, and $u_0$ is holomorphic near $0$. For each root $\tau$ of…

Consider the ramified initial-value problem … … with $N_0>2$, $c_1,c_2\in\mathbb{C}^{\times}$, and $c_j\in\mathbb{C}$ for $j\geqslant 3$. For a root $\tau$ of $\tau^2-u_x(0,0)=0$,…

In the energy-supercritical case $p=4$ and dimension $d\geq 5$, consider the singular radial solution $u_{\infty}$ of … for some $\omega_{\infty}\in(d-4,d)$ satisfying … Define the…

The $C^{p'}$ regularity conjecture. A bounded solution should satisfy

Let $1<q<\alpha<p<\gamma<p^*$, and consider the branch of weak positive solutions to problem … has an S-shaped bifurcation curve. This conjecture concerns the global structure of t…

Let $I_c$ be the variational functional associated with the stationary nonlinear Maxwell–Klein–Gordon equation, and let $v_infty$ be the global minimizer of $I_\infty$ described ab…

Let $u=u^{(0)}+u^{(1)}+\ldots+u^{(N)}$ be the order-$N$ approximative standing-wave solution obtained by the iterative procedure, and let $D_b u=0$ denote the Born–Infeld equation,…

Nonexistence conjecture. There are no nodal solutions with more than one sign change, namely

The model describes a strain field whose linearized unstable modes may be oscillatory. Suppression-and-localization conjecture. The combined effect of instability and nonlinearity…

$p$-harmonic approximation conjecture. The limiting mapping $u_\infty$ is a generalized solution of (1.1), in the sense of $D$-solutions, also in the full vectorial case.

Uniqueness conjecture. The good solution selected by the method of $L^p$ approximations as $p\to\infty$ should be unique.

Let $E^{2,2}_{\mathrm{rad}}(\mathbb{R}^3)$ be the radially symmetric subspace of the Coulomb–Sobolev space $E^{2,2}(\mathbb{R}^3)$. Ruiz's conjecture. … This concerns the endpoint…

Let $f$ be a positive continuous function on $(0,\infty)$ and consider the biharmonic inequality … in an exterior domain of $\mathbb{R}^2$. The conjecture. If … then the inequality…

Let $n\ge 3$ and let $f$ be a positive continuous function on $(0,\infty)$ satisfying … and … Consider the biharmonic inequality … in an exterior domain of $\mathbb{R}^n$. The conj…

Let $u(r)$ and $v(r)$ be the radial components of a solution of the Chern–Simons $SU(3)$ system AZ], with mixed-type behavior meaning that the two components have different asympto…

Let $D$ be the Euclidean Dirac operator on $\mathbb{R}^{n}$, and let $u$ be a solution of … Assume that $u$ has finite relative Morse index. Conjecture. Such solutions should vanis…

Consider the fourth-order $p$-Laplacian problem and the family of basic self-similar profiles … constructed in Section 5.4. Branching conjecture. Branching at $p=n+1$ is valid in a…