10 problems

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, with $N\geq3$, and let $p>1$, $\lambda>0$. Consider the Neumann problem … where $\nu$ is the outward unit normal to…

Let $\Omega\subset\mathbb{R}^n$ be a bounded uniformly convex domain. Consider the Neumann problem for the $k$-Hessian equation, with $2\leq k\leq n-1$, obtained from … with prescr…

Consider the radial Neumann problem discussed above in dimension $N=1$, with $R=1$, $p=2$, and nonlinearity $g$. For the power nonlinearity $g(s)=s^{q-1}$, the bifurcation branches…

Calanchi–Ciraolo–Messina conjecture. All classical solutions are constant for $\varepsilon$ sufficiently large. The paper's abstract states that this conjecture is proved by establ…

Boundary concentration quantization conjecture. The concentration constants are all equal to

Let $\Omega$ be a smooth and bounded domain, let $p>1$, and consider the Neumann problem referred to as equation, with parameter $\mu$. Lin–Ni conjecture. The equation has only the…

Let $(u_p)$ be a family of positive solutions of the planar elliptic Neumann problem on a bounded domain $\Omega$, satisfying the energy bound, and let $m$ denote the number of bou…

Radial Neumann solution conjecture. If

Consider the radial Neumann problem with exponent $p>2$ and power nonlinearity $g(s)=s^{q-1}$. In this case, non-constant solutions arise when $q>p$, and numerical simulations sugg…

Let $f$ and $g$ be related by … Let $u_0$ be a positive constant and let $\lambda_{k+1}^{\textnormal{rad}}$ denote the $(k+1)$-st radial Neumann eigenvalue, with $k\geq1$. BNW's os…