Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Abstract

The purpose of this article is to deal with the following biharmonic critical Choquard equation ? ? ? ? ? ? 2 u = ? f ( x ) | u | q ? 2 u + g ( x ) ( ? ? g ( y ) | u ( y ) | 2 ? ? | x ? y | ? d y ) | u | 2 ? ? ? 2 u in ? , u ,

? u = 0 on ? ? , where ? is a bounded domain in R N with smooth boundary ? ? , N ? 5 , 1 < q < 2 , 0 < ? < N , 2 ? ? = ( 2 N ? ? ) / ( N ? 4 ) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and ? > 0 is a parameter. The functions f , g : ¯¯¯¯ ? ? R are continuous sign-changing weight functions. Using the Nehari manifold and fibering map analysis, we prove the existence of two nontrivial solutions of the problem with respect to parameter ? .