On The Fu\v{c}ik Spectrum Of Non-Local Elliptic Operators

TL;DR

This paper investigates the Fučík spectrum of the fractional Laplace operator, analyzing the structure of the spectrum, properties of the first nontrivial curve, and a variational characterization of the second eigenvalue, with applications to nonresonance problems.

Contribution

It provides new insights into the Fučík spectrum for non-local fractional operators, including the existence, properties, and asymptotic behavior of spectral curves, and characterizes the second eigenvalue variationally.

Findings

Existence of a first nontrivial spectral curve with specific properties.

Lipschitz continuity and monotonicity of the spectral curve.

Variational characterization of the second eigenvalue.

Abstract

In this article, we study the Fu\v{c}ik spectrum of fractional Laplace operator which is defined as the set of all $(\al,\ba)\in \mb R^2$ such that \begin{equation*} \quad \left. \begin{array}{lr} \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\ \end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$, where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$, $s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fu\v{c}ik spectrum.