11 problems

Consider the Lane–Emden problem … where $\mathbb R^n_+=\{x_n>0\}$, $n\geq 2$, and $q>1$. Liouville conjecture. The only nonnegative solution is the trivial one, namely $u\equiv 0$.…

Let $\Omega$ be a general planar domain, without any symmetry assumption, and consider the Lane–Emden problem in which $p\to+\infty$. A cluster of bubbles is a solution whose bubbl…

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 $m\geq 1$ and let $\overset{\scriptscriptstyle{m}}{{u}}_{p}:=\overset{\scriptscriptstyle{m}}{{u}}_{0,p}$ be the radial solution of the Lane–Emden equation with $\alpha=0$ and D…

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and let $u_p$ be a family of solutions satisfying the uniform energy bound … Assume that, along a subsequence, the conce…

Let $\Omega\subseteq\mathbb{R}^2$ be a bounded domain, and let $(u_p)$ be positive solutions of the Lane–Emden problem whose energy quantity has a limit … Suppose that, along a sub…

Global branch-existence conjecture. The bifurcation branches exist for every $p\geq p^k$.

High-symmetry radiality conjecture. For $k\geq 6$ and large $p$, $u_p^k$ is radial.

Coincidence conjecture. The bifurcating solution in $\mathcal X_k$ exists for every $p\geq p^k$ and coincides with $u_p^k$ when $k=4,5$, and possibly also when $k=3$.

Gidas–Spruck conjecture. There is no positive solution for all