13 problems

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

Let $p\in[1,2)$ and let $\Omega\subset\mathbb{R}^N$ be a smooth open bounded domain. Assume that $\Omega$ is connected and that the sublinear Dirichlet problem considered in the so…

Let $f$ be the nonlinearity in the radial equation … that has precisely $k-1$ nodes. This conjecture concerns the nodal characterization of bound states and asserts uniqueness with…

Esposito–Musso–Pistoia conjecture. For all sufficiently large $p$, the family $v_p$ is a family of least energy nodal solutions. The preceding corollary proves the forward implicat…

Grossi–Grumiau–Pacella conjecture. The displayed boundedness condition should hold automatically for least energy nodal solutions. This condition is relevant because, when it holds…

Let $\Omega\subset\mathbb R^2$ be a square, and let $p$ be close to $2$. A least energy nodal solution (l.e.n.s.) is a sign-changing solution with minimum energy. Symmetry conjectu…

Let $n=6\ell$ with $\ell\in\mathbb{N}$. In these dimensions, consider structures given by the doubling of the equatorial from Theorem. Maximal-rank conjecture. Combining $2\ell$ st…

Let $n$ be an odd positive integer. The doubling of the equatorial in three coordinates, together with desingularizations of equators in two or four coordinates, is considered as a…

Let $\widetilde u_p^k$ denote the nodal least-energy solution in the $k$-invariant space $H^1_{0,k}$, and let the corresponding solutions found by bifurcation be $k$-invariant. The…

Let $\Omega$ be a general bounded domain, and let $\bar\lambda_2=\bar\lambda_2(N,\Omega)$ be a limit value of the parameter. The preceding results concern nodal radial solutions wi…

Let $\Omega\subset\mathbb{R}^N$ and consider the Hénon-Lane-Emden system … where $p,q$ satisfy the subcriticality condition in the source and $\alpha,\beta\geq 0$. A least energy n…

Closed-geodesic balanced-pattern conjecture. A possible balanced pattern may occur for spikes tightly aligned near a closed geodesic of $\partial\Omega$.

Alternate-sign peak alignment conjectures. (C1) There exists a solution with alternate-sign peaks aligned on an interior straight line intersecting $\partial\Omega$ orthogonally. (…