4 problems

Let $T>0$, let $a:[0,T]\to\mathbb{R}$ be a two-step indefinite weight, and let $\gamma\in\mathbb{R}\setminus\{-1,0,1\}$. The conjecture asserts that the boundary-value problem…

Does there exist a sequence of solutions to the self-dual Chern–Simons–Higgs equation exhibiting genuinely non-simple blow-up at a single concentration point; that is, can at least…

For every speed $c\in(0,\sqrt{2})$, there exists a nonconstant finite-energy profile $\psi:\mathbb{R}^2\to\mathbb{C}$ such that the traveling wave $\Psi(t,x)=\psi(x-ct e_1)$ solves…

Let $(M^n,g)$ be a closed Riemannian manifold with $\operatorname{Ric}_g\geq (n-1)g$. For $1<p<n$, $p<q<p^*:=\frac{np}{n-p}$, and $\lambda>0$, determine when every positive solutio…