An existence result for fractional problems with mixed boundary conditions

In this paper we investigate the existence and qualitative properties of weak solutions for a class of nonlinear fractional equations driven by the spectral fractional Laplacian under mixed Dirichlet--Neumann boundary conditions. More precisely, we consider subcritical problems of the form \begin{equation*} \left\{ \begin{array}{ll}
(-\Delta)^s u=\lambda u + \mu f(x,u) & \text{ in } \Omega\smallskip, \\ \displaystyle u\chi_{\SD} +\frac{\partial u}{\partial\nu}\chi_{\SN}=0 & \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where \(s\in(1/2,1)\), \(\Omega\subset\mathbb R^N\) is a bounded smooth domain, and the boundary \(\partial\Omega\) is endowed with a mixed Dirichlet--Neumann configuration satisfying suitable geometric and analytic assumptions. Working in the natural spectral fractional framework associated with the mixed boundary operator, we establish the existence of nontrivial weak solutions by means of suitable variational methods. The analysis is based on a topological and variational principle combined with a localization argument involving certain compactly supported cutoff functions. More precisely, the proof relies on a delicate local comparison argument involving localized plateau-type test functions and explicit asymptotic estimates near the origin. Such an approach is sufficiently flexible and may potentially be adapted to several different settings. As applications, we establish simplified existence criteria in the autonomous case through the use of the Chebyshev radius of the domain and present concrete examples illustrating the applicability of the abstract theory.