28 problems

Let $N\ge 3$, let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$, and let $0\in\partial\mathbb{R}^N_+$. Consider positive solutions of the critical problem … with an isolated singular…

Uniqueness conjecture. The problem admits at most one solution.

Uniqueness conjecture. If $\alpha=4\pi$ and $\Omega$ is a ball, then the maximizers of the Trudinger–Moser inequality are unique.

Stability conjecture. If $n\leq 7$, then all level sets of $u$ are hyperplanes.

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 $\Omega$…

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…

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…

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 $x\in\Omega$,…

Let $\Omega$ be the domain and let $u^{\varepsilon}$ and $u$ denote, respectively, the solution of the rapidly oscillating elliptic equation and its homogenized limit, with data…

Let $\mathcal L\in\operatorname{\mathcal S\mathcal E}$, and let $G\subset\mathbb C$ be a bounded simply connected domain. The domain $G$ is $\mathcal L$-regular when the associated…

Let $\mathcal L\in\operatorname{\mathcal S\mathcal E}$, and let $X$ be a compact set in $\mathbb C$. The spaces $P_{\mathcal L}(X)$ and $C_{\mathcal L}(X)$ denote, respectively, th…

Monotone-profile conjecture. There exist $\gamma,m>0$ such that

Uniqueness conjecture. There exist $\gamma,m,z\in\mathbb{R}$ such that

Let $\Lambda^*<\lambda_*<\Lambda_1$ and set $\beta_*=1$. Let $S_{1,\lambda,\beta}$ denote the class of radial sign-changing solutions with the index specified in the paper, and let…

Regularity and minimal-growth solution conjecture. Then:

“Qualitative” Landis conjecture. Then $u\equiv 0$.

Higher-dimensional convergence conjecture. The sequence $cv_n$ is bounded in $W_0^{1,2}(\Omega)\cap L^\infty(\Omega)$ and therefore converges, up to subsequences, to a bounded nonn…

Let $cOmega$ be a bounded domain, let $comegasubsetsubsetcOmega$ be the region where the nonnegative datum is supported, and let $u$ be the function given by Theorem 1 with…

Assume (A1): $\beta$ is non-negative and $C^1$, with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$. Let … where $\Phi'=\beta$. Dimension-seven non-one-dimensional minimizer co…

Assume (A1)–(A2): $\beta$ is non-negative and $C^1$ with $\textrm{supp}\,\beta=[0,1]$ and $\int_0^1\beta=1$, while $u\geq0$ satisfies $\|\nabla u\|_{L^\infty(\mathbb{R}^n)}<\infty$…

Let $(\lambda(\alpha),U(x;\alpha))$ and the constants $\alpha_1$, $\alpha_2$, and $\alpha_3$ be as in Theorem, where $\alpha_*<\alpha_1\leq\alpha_2\leq\alpha_3$ describe the degene…

Let $\Omega\subset\mathbb{R}^n$ be a ball, and consider the Trudinger–Moser variational problem … A maximizer is a function attaining this supremum. Uniqueness conjecture. Maximize…

Let $(\beta_1,\ldots,\beta_{k-1})$ be the configuration appearing in the $k$-layer limit problem of Theorem defining the corresponding multi-layer radial solution. Uniqueness conje…