An Ambrosetti-Prodi type result for fractional spectral problems

TL;DR

This paper extends Ambrosetti-Prodi type results to fractional spectral problems, showing the existence of multiple solutions for certain nonlinear fractional PDEs using variational methods.

Contribution

It establishes the existence of multiple solutions for fractional spectral problems with Ambrosetti-Prodi type nonlinearities, including periodic cases, using variational techniques.

Findings

Existence of at least two solutions for the problem when parameter t is below a threshold.

Identification of a critical parameter t_0 where solution multiplicity occurs.

Extension of classical results to fractional Laplacian operators and periodic settings.

Abstract

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right. \end{equation*} where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $s\in (0, 1)$, $N> 2s$, $(-\Delta_{Dir})^{s}$ is the fractional Dirichlet Laplacian, $f: \bar{\Omega} \times \mathbb{R} \rightarrow \mathbb{R}$ is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti-Prodi type assumptions, $t\in \mathbb{R}$, $\varphi_{1}$ is the first eigenfunction of the Laplacian with homogenous boundary conditions, and $h:\Omega\rightarrow \mathbb{R}$ is a bounded function. Using variational methods, we prove that there exists a $t_{0}\in \mathbb{R}$ such that the above problem admits at least two distinct solutions for any $t\leq t_{0}$. We also discuss the existence of solutions for a fractional periodic Ambrosetti-Prodi type problem.