5 problems

Let $a\in\mathcal{C}([0,L])$ satisfy the sign condition that it is positive on $n$ separated intervals $I_i^+=(\sigma_i,\tau_i)$ and negative on the intervening intervals, together…

Extension conjecture. Theorem $7.4$ remains true if $a(x)$ satisfies $(a_5)$ instead of $(a_6)$; namely, under $(f_2)$, $(a_2)$, and $(6.18)$, problem $(1.1)$ cannot admit regular…

Generic asymmetric bifurcation conjecture. Generically, the set of positive solutions consists of $C^+$ together with $n$ supercritical folds $D_j$, $j\in\{1,\ldots,n\}$, and, as…

Gómez-Reñasco–López-Gómez conjecture. There exists $lambda_c<pi^2$ such that, for every $lambda<lambda_c$, the problem admits at least $2^{n+1}-1$ positive solutions.

Consider a superlinear indefinite problem with homogeneous Dirichlet boundary conditions, parameter $\lambda$ sufficiently negative, and a weight function $a(t)$ having $n e 0$ com…