42 problems

Let $\Omega$ be a Lipschitz domain, and let $u\in H^1$ be a weak solution to the Dirichlet problem … Suppose that the set…

Let $(\Omega,\varrho)$ be a domain and boundary parameter for the problem … lambda(Omega,varrho)=f'(0), … has no positive bounded solution. This extends the established conclusion…

Let $(\Omega,\varrho)$ be a domain with the boundary conditions and elliptic problem specified by … admits at most one positive bounded solution. The paper's methods do not treat c…

Branch-shape and uniqueness conjecture. The solution branches in cases VII2-1 and VII2-2 of Theorem VII1 have the same behaviour as the branches depicted in the case-IIb and case-V…

Multiplicity and branch-shape conjecture. The branches of solutions behave as depicted in the case-VI bifurcation diagram: the problem has at least two solutions for every…

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…

Multiplicity conjecture. For each $M\in\mathbb{N}_0$, there exists at least one distinct solution $u_M$ of problem $$ having at most $M$ modes.

Let $f$ be a nonlinearity in the smooth asymptotic class of the dimensional-bound theorem, let $T_f$ denote its blow-up level, and define … For a corresponding stable solution…

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…

Let $B\subset\mathbb{R}^{n}$ be a unit ball with $n\geq 3$. Let $u:4B\to\mathbb{R}$ be harmonic and satisfy $u(0)=0$, and let $N_u(0,\frac{1}{2})$ denote its doubling index at the…

Berestycki–Caffarelli–Nirenberg conjecture. (i) If there is a positive bounded solution $u$ of this problem, then $u$ depends only on $x_n$; consequently, $u_{x_n}>0$ and…

Let $\Omega$ be either $\mathbb{R}^N$ or an exterior domain in $\mathbb{R}^N$, and let $V\in L^{\infty}(\mathbb{R}^N)$ satisfy $\|V\|_{\infty}\leq 1$. Suppose that $u$ solves … and…

Let $B$ be the unit ball of $\mathbb{R}^n$, and let $k\in\mathbb{N}$ satisfy $n>2k\geq 2$. Consider $\lambda\in\mathbb{R}$ and $u\in C^{2k}(\overline{B})$ solving … where…

Let $T$ be a triangle with mixed boundary conditions: Neumann conditions on two sides and a Dirichlet condition on the remaining side. Consider the first positive non-trivial eigen…

Let $\Gamma$ be a smooth surface, and let $\mathcal I+\mathcal R_{\text{LB}}$ denote the integral operator in, where $\mathcal R_{\text{LB}}$ is the composition of the Laplace–Belt…

Guo–Wang's conjecture. If $1<q<\frac{n}{n-2}$ and $0<\lambda\leq\frac{1}{q-1}$, then $u$ is constant.

Let $m\in\mathbb{N}$ and let $\Omega_k$, $k\in\{1,\dots,m\}$, be as in the introduction. Consider the multi-phase overdetermined problem described by problem … admits a solution…

Let $\phi: \mathbb{R}^{n-1} \times \mathbb{R}^+ \to \mathbb{R}$. Suppose that $\phi(s,0) \equiv 0$. Let $f: \mathbb{R}^{n-1} \to \mathbb{R}$, and suppose … Here $g$ is the one-dime…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$. The concave-solution conjecture. If $u$ is concave, then $u$ is one-dimensional. The paper proves the analogous…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$, and let $\delta$ be the standard Euclidean metric. Assume that $\mathbb{R}^2$ has diameter $pi$ under the metric…

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain and let $\Sigma=B\cap\partial\Omega$ for a ball $B$ centered at a point of $\partial\Omega$. Let $u$ be harmonic in $\Omega$,…

Braverman–Milatovic–Shubin conjecture. If $(M,g)$ is a geodesically complete Riemannian manifold, then the $L^2$-positivity preserving property holds.

Non-degeneracy conjecture. Except for finitely many $\tau$ modulo the $SL(2,\mathbb{Z})$ action, every solution of the Toda system is non-degenerate. If true, this would support th…

Let $u$ be a global solution of an equation of minimal-surface type on $\mathbb{R}^n$, corresponding to a functional $A_{\Psi}$. Here $B_r$ denotes the radius-$r$ ball, $\nabla u$…

Consider the viscous Hamilton-Jacobi equation … where $\Omega$ is a domain, $\gamma>1$, and $f\in L^q(\Omega)$ with $q>1$. Write $\gamma'=\frac{\gamma}{\gamma-1}$ for the conjugate…