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Asymptotically linear fractional problems with mixed boundary conditions

Giovanni Molica Bisci, Alejandro Ortega, Luca Vilasi

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.

Asymptotically linear fractional problems with mixed 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 is odd and a suitable relation between the perturbation parameter, the limit of as and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.
Paper Structure (3 sections, 3 theorems, 60 equations)

This paper contains 3 sections, 3 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Functional framework and pseudo-index theory
  3. Proof of Theorems \ref{['mainresult']} and \ref{['thmsubcritgrowth']}

Key Result

Theorem 1

Assume $(\Omega_1)-(\Omega_3)$. Then,

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['mainresult']}
  • proof : Proof of Theorem \ref{['thmsubcritgrowth']}
  • Remark 4
  • Remark 5
  • Remark 6