33 problems

Lane–Emden conjecture. If $(p,q)$ is subcritical, then the system admits no positive solution in $\left(C^2(\mathbb{R}^N)\right)^2$.

Lazer–McKenna conjecture. There should be at least $2k$ solutions of the boundary-value problem.

Hénon–Lane–Emden conjecture. If $(p,q,a,b)$ is subcritical, then the system admits no positive solutions in $\left(C^2(\mathbb{R}^N\setminus\{0\})\cap C(\mathbb{R}^N)\right)^2$.

Metric-measure and Heisenberg-group conjecture. Problem $$ should yield the same result when considered over a metric measure space or a Heisenberg group.

Variable-exponent conjecture. Theorems concerning multiplicity for … should remain valid when $q$ is replaced by a variable exponent $q$.

Let $N=2$ and let $m>1$. For each positive mass $\mu>0$, consider the polyharmonic Dirichlet and Navier boundary-value problems under assumptions (FP1)--(FP4), with the normalizati…

Let the curves in Figure be energy curves for the problem, with solutions parametrized by their energy and with parameter $\lambda\in\mathbb{R}$. The conjectured joined diagram is…

de Figueiredo–Lions–Nussbaum conjecture. This condition is not necessary for the uniform a priori bounds established for positive solutions, although it is essential in their proof…

Uniqueness conjecture. Solutions to in $\mathcal M$ satisfying the displayed pointwise bound are unique.

Let $N\geq 1$, $\gamma>0$, $\mu>0$, $c>0$, and $2+\frac{4}{N}<q<2^*$. Let $\omega_1$ and $\omega_2$ be the parameters appearing in the condition governing the existence of normaliz…

Let problem … possesses $2^{\kappa+1}-1$ positive solutions. The conjecture is motivated by the almost-everywhere convergence of positive solutions to zero on the zero set of the w…

The defocusing Gross–Pitaevskii–Poisson equation admits mountain pass solutions of type I and type II for the same mass parameters, with the two types defined by the respective con…

Consider the interior problem, in which the set $K$ contains the positivity set $\Omega$ of a solution $u$, so that $K\supset\Omega$. The sub-level sets are the sets $\{u<l\}$ for…

The paper studies a free boundary problem for a nonnegative solution $u$, with positivity set $\Omega=\{u>0\}$, in the convex setting. Uniqueness conjecture. The solution to the fr…

Let $\Lambda_1$ and $\Lambda_2$ be the trace spaces associated with the two subdomains $\Omega_1$ and $\Omega_2$, and let $\Lambda$ be the common interface trace space defined in t…

Takáč's conjecture. The assumption that $\partial\Omega$ is connected can be omitted for sufficiently regular domains.

Asymptotic energy comparison conjecture. In view of the dependence of the limiting energies on the nonlinearities, one expects

Consider positive solutions of … under the condition $f_r(r,u)\leq 0$. In dimension $n=1$, the value $u(0)$ is known to be a global parameter. Global-parameter conjecture. The same…

Let ${\mathfrak m}$ be the coefficient arising in the expansion of the squared $L^2$ norm of the concentrating profile, and suppose that the condition … {mathfrak m}0 … ; its valid…

Let $\rho$to$2\sigma_0$ from below or above, and consider the Dirichlet and Neumann problems on a ball, respectively, with single-peak solutions concentrating at the origin. Here,…

Consider the planar dynamical system associated with the radial solutions of the scaling-invariant nonlinear elliptic equation, with dimension parameter $N=2$. A cycle means a peri…

Assume $N\geq 3$, let $M<0$, and consider positive radial solutions $u$ with the asymptotic profile $U_M(r)$ as $r\to0$. A solution has the far-field behavior…

Consider the Lane–Emden system and its least energy solutions $(u_{\epsilon},v_{\epsilon})$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$, with exponent $p$ and the associ…

Let $q>1$, let $\sigma$ be a nonnegative measure satisfying the hypotheses of the paper, and let $K\subset\Omega$ be compact. A compact set $K$ is called removable for the relevant…

Let $c_{TF}>0$, and for $0<\mu<\frac{15}{64c_{TF}}$ let $Q_\mu$ be the unique positive solution of the associated Euler–Lagrange equation, with mass $M(\mu)$, and let…