15 problems

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

Let $M$ be a metric-measure space, let $up1$, and let $Omegasubset M$ be a connected open set containing a fixed origin $o$. Write $r(x)=\operatorname{dist}(x,o)$, and let…

Complete classification conjecture. Either $u\equiv 0$, or there exist $\lambda:=u(0)^{\frac{p}{N-p}}>0$ and $x_{0}\in\mathbb{R}^{N}$ such that

Let $d\geq 2$, let $\nabla_A$ denote the magnetic gradient, let $h_{A,p}$ be the corresponding magnetic $p$-Laplacian form domain, and let $\mu_{p,d}$ be the Hardy constant. Let…

Regularity and minimal-growth solution conjecture. Then:

Let $1<p<\infty$, and let $\Delta_p$ denote the $p$-Laplacian of the oriented hypergraph under consideration. An eigenvalue is called a min-max eigenvalue if it is expressible in t…

Let $M$ be a closed $n$-dimensional Riemannian manifold, let $p$ be the exponent in the $p$-Laplacian, and let $c_n(p)$ be the universal constant in the asymptotic law … Here…

Let $\Omega$ be the domain, let $f=f^+-f^-$ be the prescribed source term, and let $(\mu(t),u(t))$ solve the extended dynamic Monge–Kantorovich equations with initial datum $\mu_0$…

Existence conjecture. There exists $1<\underline{m}<(p^*)'$ such that, if $f\in L^m(\Omega)$ with $m\geq\underline{m}$, then system $$ admits a weak solution $(u,\varphi)\in W^{1,p…

Liouville-type conjecture. Every weak solution $u\in\mathcal D^{1,p}_0({\mathbb R}_+^N)$ of this equation is identically zero:

Generalized eigenvalue quotient conjecture. If $q<p$, then functionals of the form $\mathcal{R}_{p,q}(\Omega)$ admit a minimizer and are bounded above. Any sequence of fixed-volume…

Let $Q'(u)=0$ be the p-Laplacian type equation in a domain $\Omega$, let $\zeta\in\{0,\infty\}$ be an isolated singular point, and let $\mathcal{G}_\zeta$ denote the germs of posit…

Strong comparison principle. Either $u<v$ throughout $\Omega'$, or $u=v$ throughout $\Omega'$. The weak comparison principle supplies the non-strict inequality $u\leq v$; the conje…

Let $X$) be the domain in Eq. (1), let $\hat X$ be its compactification, and let $\zeta\in\partial\hat X$. Assume that Eq. (1) has a Fuchsian type singularity at $\zeta$ and admits…

Let $1<p<\infty$, let $Q_V$ be the functional under consideration on a domain $\Omega$, and let $\Omega_1\Subset\Omega$ be a smooth open set. Denote by…