The existence of nontrivial solution for a class of sublinear biharmonic equations with steep potential well

where 2u = ( u), N > 4, λ > 0, 1 < q < 2 and μ ∈ [0,μ0], 0 < μ0 <∞. The continuous function f verifies the assumptions: (f1) f (s) = o(|s|) as s→ 0; (f2) f (s) = o(|s|) as |s| →∞; (f3) F(u0) > 0 for some u0 > 0, where F(u) = ∫ u 0 f (t) dt. According to hypotheses (f1)–(f3), the number cf = max s =0 | f (s) s | > 0 is well defined (see [1]). The continuous functions α and K verify the assumptions: (α1) 0 < α(x) ∈ L1(RN )∩ L∞(RN ) and cf ‖α‖∞ < 1; (K1) 0 < K(x) ∈ L 2 2–q (RN )∩ L∞(RN ). We require the potential V :RN →R to satisfy the following assumptions: (V1) V (x) is a nonnegative continuous function onRN , there exists a constant c0 > 0 such that the set {V < c0} := {x ∈RN |V (x) < c0} has finite positive Lebesgue measure;