S. Buccheri | Searcharxiv

arXiv2024

Elliptic problems with superlinear convection terms

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type \[ -\mbox{div}(M(x)\nabla u)= -\mbox{div}(h(u)E(x))+f(x), \] where $M$ is a bounded elliptic matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces, and the function $s\to h(s)$ features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle.

math.AP↗

arXiv2021

A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to\infty$

In this work, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $λ_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{ \begin{array}{rclcl} -Δ_p u + (-Δ)^r_p u & = & \frac{2α}{α+β}λ|u|^{α-2}|v|^βu & \mbox{in} & Ω\\ -Δ_p v + (-Δ)^s_p v& = & \frac{2β}{α+β}λ|u|^α|v|^{β-2}v & \mbox{in} & Ω u& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω v& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation} where $Ω$$\subset$ $\mathbb{R}^N$ is a bounded open domain, $0<r, s<1$ and $α(p)+β(p) = p$. Moreover, we address the asymptotic limit as $p \to \infty$, proving the explicit geometric characterization of the corresponding first $\infty-$eigenvalue, namely $λ_{\infty}$, and the uniformly convergence of the pair $(u_p,v_p)$ to the $\infty-$eigenvector $(u_{\infty},v_{\infty})$. Finally, the triple $(u_{\infty},v_{\infty},λ_{\infty})$ verifies, in the viscosity sense, a limiting PDE system.

math.AP↗