184 problems

Let $u\in C^2(\mathbb{R}^n)$ be a bounded entire solution of … where $W:\mathbb{R}\to\mathbb{R}$ is a double-well potential satisfying the stated regularity and sign conditions, an…

Let $u:\mathbb{R}^n\to\mathbb{R}$ be a bounded solution with … uniformly with respect to $x'\in\mathbb{R}^{n-1}$. A solution is one-dimensional when it depends on only one spatial…

Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the functional … Assume $1<p<\frac{n+2-k}{n-2+k}$ for…

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 $u\equiv 0$.…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, with $N\geq3$, and let $p>1$, $\lambda>0$. Consider the Neumann problem … where $\nu$ is the outward unit normal to…

A divergence-form elliptic operator has bounded measurable coefficients satisfying a Carleson measure condition of the type arising from changes of variables in smooth elliptic or…

Let $\Omega$ be a domain and let $u$ solve the semilinear Dirichlet problem … Here, $u$ is quasiconcave when its superlevel sets are convex. Lions' conjecture. For a general $f$, t…

Lions's conjecture. For every $p>\frac{n}{r}$, an $L^p$ estimate should hold for $D^2u$ whenever $f\in L^p$.

Let $f\in C^1(\mathbb R)$ satisfy the positivity and Keller–Osserman conditions, and let $u$ be a solution of the boundary blow-up problem studied in the source on the unit ball…

Let $\Omega$ be a domain for which the limit problem admits the second solution described in the paper, and let the corresponding finite-$p$ problem have mountain-pass solutions. M…

Coincidence conjecture. If $\Omega$ is a convex $C^{1,1}$ domain, then

Sobolev hyperbola conjecture. The Sobolev hyperbola is the critical dividing curve between existence and nonexistence of solutions to the Hardy–Hénon system. This conjecture propos…

Let $u$ be a smooth function on the standard sphere $\mathbb{S}^{2}$, and let ${\Greekmath 010B}$ be a positive constant satisfying … A solution is axially symmetric if it is invar…

Lin's axial symmetry conjecture. If $\gamma>0$ and $\beta\leq 16\pi$, then every solution is axially symmetric with respect to $\vec n$.

Lin–Wang's nonexistence conjecture. For every $n\geq2$, this equation has no solutions when $\tau\in i\mathbb{R}_{>0}$.

Quasi-metric inequality conjecture. In this setting, the inequality holds, with the obvious modifications, for bounded Lipschitz domains $\Omega$ and operators $L$ with bounded mea…

Let $L$ be a differential operator with bounded measurable coefficients on a domain $\Omega$, and let $G$ be its Green function. Suppose the associated transformed kernel has the q…

Let $D$ be a bounded domain in ${\mathbb R}^n$ with boundary of class $C^{1/p}$, where $0<p\leq 1$. Let $G(f)$ and $N(f)$ denote the solutions of the Dirichlet and Neumann problems…

For $p>2$, consider the exponent $m$ arising in the measure-data problem for the $p$-Laplace equation; in the paper's notation, $m$ is obtained by multiplying the corresponding exp…

Let $\Omega$ be a convex bounded domain in $\mathbb{R}^3$ with smooth boundary. For a sequence $w_{p_n}$ associated with $p_n\to\infty$, let $S$ denote the blow-up set of a subsequ…

Extremality conjecture. In the class of operators defined by the ellipticity condition in, all critical exponents lie in the same interval; equivalently, the operators…

Let $(S)_\lambda$ be the stationary electrostatic MEMS problem on a domain $\Omega\subset\mathbb{R}^N$, with critical voltage $\lambda^*$ and positive permittivity profile $f$. Pos…

Let $\Omega$ be the domain, let $f$ be the permittivity profile on $\Omega$, let $\lambda^*$ be the pull-in voltage, and let $\phi_\Omega$ be the first eigenfunction of the relevan…

Let $(S)_{\lambda}$ denote the stationary electrostatic MEMS problem on a domain $\Omega\subset\mathbb{R}^N$, with critical voltage $\lambda^*$ and permittivity profile $f$. Nonexi…

Let $B(0,1)\subset\mathbb{R}^d$ be the unit ball. Let $\lambda\in C^\infty(\overline{B(0,1)})$ satisfy $\lambda>0$ on $\overline{B(0,1)}$, and let…