10 problems

Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the functional … Assume $1<p<\frac{n+2-k}{n-2+k}$ for…

Let $u_\alpha$ be a ground state solution of problem, and let $V_\alpha$ be the associated potential. The maximum of $V_\alpha$ is attained at $1/2$. Concentration conjecture. Grou…

Positivity conjecture for $\alpha_N$. The constant $\alpha_N$ is positive for every $N$.

Peak-configuration conjecture. The possible configuration of peaks is

Ni's conjecture. For every integer $1\leq\mathbf k\leq\mathbf d-1$, there exists $p_{\mathbf k}\in(1,\infty)$ such that, for every $1<p<p_{\mathbf k}$, the problem has a solution…

Bahri–Pistoia conjecture. Positive nonradial solutions in general domains should exhibit similar asymptotic behavior as $\varepsilon\to 0$, namely concentration and the correspondi…

Magnetic concentration generality conjecture. The influence of the magnetic field on the location of concentration, including dependence on the tangential component of $A$ and on…

Concentration-set conjecture. Problem $$ actually possesses solutions which have $m$-dimensional concentration sets for every $0leq mleq N-1$.

Consider the Neumann problem … with … where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $\mathbf{n}$ is the inward normal to $\partial\Omega$, $\varepsilon$ is small, an…

Let $\mathcal D$ be the domain under consideration, let $\Gamma$ be a nondegenerate geodesic of its boundary with inner normal curvature, and let $p$ approach the first Sobolev cri…