Asymptotically linear fractional problems with mixed boundary conditions

TL;DR

This paper proves the existence and multiplicity of solutions for an asymptotically linear spectral fractional Laplacian problem with mixed boundary conditions, using variational methods and pseudo-index theory.

Contribution

It introduces new existence and multiplicity results for fractional Laplacian equations with mixed boundary conditions, extending previous work to asymptotically linear nonlinearities.

Findings

Existence of solutions under certain spectral conditions.

Multiplicity results via pseudo-index theory.

Application to problems with mixed Dirichlet-Neumann boundary conditions.

Abstract

We derive the existence of solutions for an asymptotically linear equation driven by the spectral fractional Laplacian operator with mixed Dirichlet-Neumann boundary conditions. When the nonlinear term $f$ is odd and a suitable relation between the perturbation parameter, the limit of $f(\cdot,t)/t$ as $t\to 0$ and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.