Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Some Extremal Problems for Hereditary Properties of Graphs

Given an infinite hereditary property of graphs $\mathcal{P}$, the principal extremal parameter of $\mathcal{P}$ is the value\[ \pi\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}\binom{n}{2}^{-1}\max\{e\left( G\right) :\text{ }G\in\mathcal{P}\text{ and }v\left(G\right) =n\}.\]The Erdős-Stone theorem gives $\pi\left( \mathcal{P}\right) $ if $\mathcal{P}$ is monotone, but this result does not apply to hereditary $\mathcal{P}$. Thus, one of the results of this note is to establish $\pi\left( \mathcal{P}\right) $ for any hereditary property $\mathcal{P}.$Similar questions are studied for the parameter $\lambda^{\left( p\right)}\left( G\right)$, defined for every real number $p\geq1$ and every graph $G$ of order $n$ as\[\lambda^{\left( p\right) }\left( G\right) =\max_{\left\vert x_{1}\right\vert^{p}\text{ }+\text{ }\cdots\text{ }+\text{ }\left\vert x_{n}\right\vert ^{p} \text{ }=\text{ }1}2\sum_{\{u,v\}\in E\left( G\right) }x_{u}x_{v}.\]It is shown that the limit\[ \lambda^{\left( p\right) }\left( \mathcal{P}\right) =\lim_{n\rightarrow\infty}n^{2/p-2}\max\{\lambda^{\left( p\right) }\left( G\right) :\text{ }G\in \mathcal{P}\text{ and }v\left( G\right) =n\}\]exists for every hereditary property $\mathcal{P}$.A key result of the note is the equality \[\lambda^{(p)}\left( \mathcal{P}\right) =\pi\left( \mathcal{P}\right) ,\]which holds for all $p>1.$ In particular, edge extremal problems andspectral extremal problems for graphs are asymptotically equivalent.