Nonlocal eigenvalue problems with indefinite weight

Abstract

In the present paper, we consider a class of eigenvalue problems driven by a nonlocal integro-di erential operator \scrL K with Dirichlet boundary conditions. Under certain assumptions on p and q, we establish that any \lambda > 0 su ciently small is an eigenvalue of the nonhomogeneous nonlocal problem (\scrP \lambda ). ®§£«ï¤ õâìáï a« á á ̄¥aâà «ì­ ̈å § ¤ ç, ̄®¢'ï§ ­ ̈å ÷§ ­¥«®a «ì­ ̈¬ ÷­â¥£à®¤ ̈ä¥à¥­æ÷ «ì­ ̈¬ ® ̄¥à â®à®¬ \scrL K ÷§ aà ©®¢®î 㬮¢®î „ ̈à ̈å«¥. ‡ ̄¥¢­ ̈å ̄à ̈ ̄ã饭ì 鮤® p ÷ q ¤®¢¥¤¥­®, é® a®¦­¥ ¤®áâ ­ì® ¬ «¥ \lambda > 0 õ ¢« á­ ̈¬ §­ 祭­ï¬ ­¥®¤­®à÷¤­®ù ­¥«®a «ì­®ù § ¤ ç÷ (\scrP \lambda ).