Principal curves to fractional mm-Laplacian systems and related maximum and comparison principles

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinar systems involving fractional $m$-Laplacian operators.Comment: 18 page

Fractional elliptic systems with nonlinearities of arbitrary growth

In this article we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: \displaylines{ \mathcal{A}^s u= v^p \quad\text{in }\Omega\cr \mathcal{A}^s v = f(u) \quad\text{in }\Omega\cr u= v=0 \quad\text{on }\partial\Omega } where $s\in (0, 1)$ and $\mathcal{A}^s$ denote spectral fractional Laplace operators. We assume that $1< p<\frac{2s}{n-2s}$, and the function f is superlinear and with no growth restriction (for example $f(r)=re^r$); thus the system has a nontrivial solution. Another important example is given by $f(r)=r^q$. In this case, we prove that such a system admits at least one positive solution for a certain set of the couple (p,q) below the critical hyperbol

Least energy solutions for affine pp-Laplace equations involving subcritical and critical nonlinearities

The paper is concerned with Lane-Emden and Brezis-Nirenberg problems for a nonlocal quasilinear operator $\Delta_p^{\cal A}$ coming from convex geometry, called affine $p$-Laplace operator, which coincides with the corresponding classical one on balls for radial functions. More specifically, $\Delta_p^{\cal A}$ has been introduced in \cite{HJM5} driven by the affine $L^p$ energy ${\cal E}_{p,\Omega}$ due to Lutwak, Yang and Zhang \cite{LYZ2}. We are particularly interested in positive $C^1$ solutions of least energy type. The existence is achieved via direct method to functionals with non-coercive geometry. Key ingredients like compactness in $L^q(\Omega)$ of affine balls in $W^{1,p}_0(\Omega)$, established by Tintarev \cite{T1}, and weak lower semicontinuity on $W^{1,p}_0(\Omega)$ of ${\cal E}_{p,\Omega}$, among others, play a fundamental role in our approach. Nonexistence results are obtained by means of integral arguments.Comment: Comments are welcome