The existence of nontrivial solution for biharmonic equation with sign‐changing potential

In this paper, we study the following biharmonic equation: B vΔ 2 u + V λ ( x ) u = α ( x ) f ( u ) + ν K ( x ) | u | q − 2 u inR N ,u ∈ H 2 ( R N ) ,where N ⩾5, ν  ∈ (0, ν 0 ],1  <   q   <  2,Δ 2 u   =  Δ(Δ u ) and V λ ( x )  =   λ a ( x ) −  b ( x ) with λ  > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem B ν .