Three non-zero solutions of a Neumann eigenvalue problems involving the fractional p-Laplacian
Somnath Gandal
TL;DR
This work studies the multiplicity of nonzero weak solutions for a nonlocal Neumann problem involving the fractional $p$-Laplacian $(-\\Delta)^{s}_{p}$ on a bounded domain $\\Omega$. It adopts a variational approach in the space $W^{s,p}_{\\Omega}$ with energy $J_{\\lambda}$ and applies two critical-point frameworks to establish three-solution results in two regimes: (i) $N<sp$ and (ii) $N\\geq sp$, under subcritical growth and Carath\\odory-type conditions on the nonlinearity $h$. In each regime, explicit intervals for the parameter $\\lambda$ are derived, depending on Sobolev constants and the data $a$ and $H$, guaranteeing at least three weak solutions. This extends local multiplicity results to fractional nonlocal Neumann problems, providing multiple steady states for nonlocal diffusion models with Neumann-type boundary interactions and potential biological/physical applications.
Abstract
In the present paper, we establish a multiplicity result for a following class of nonlocal Neumann eigenvalue problems involving the fractional p-Laplacian. \begin{align} \begin{cases} (-Δ)^{s}_{p}u + a(x) \abs{u}^{p-2}u =λh(x,u) & \text {in } Ω, \mathcal{N}_{s,p}u=0 & \text {in } \mathbb{R}^N \setminus \overlineΩ, \end{cases} \end{align} Precisely, we demonstrate the existence of an open interval for positive eigenvalues $λ$, for which the problem has at least three non-zero solutions in $W^{s,p}_Ω.$