8 problems

For the hyperbolic Hénon-type equation with exponential nonlinearity studied by Huang and Zhao, is every radial solution stable whenever $n\ge 10+4\alpha$ (with $\alpha>-2$)?

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

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 $q$.

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

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…

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…

Uniqueness conjecture. Then $\lambda_1=\lambda_2$ and $r_2=r_1(\cdot+\xi)$ for some $xi\in\mathbb{R}^3$.