Abstract
Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.
Highlights
Assume that F is even in u and the following conditions hold: (F1)
Introduction and Main ResultsConsider the fourth-order Navier boundary value problemΔ2u + cΔu + a (x) u = f (x, u), in Ω, (1)u = Δu = 0, on ∂Ω, where Ω ⊂ RN (N > 4) is a bounded smooth domain, a ∈ L∞(Ω), c ∈ R, and f ∈ C(Ω × R, R)
uun−n → 0, as n → ∞. It follows from the equivalence of the norms on the finite dimensional space E0 that there exists K1 > 0 such that
Summary
Assume that F is even in u and the following conditions hold: (F1) Assume that F satisfies (F1) and (F4) there exist three positive constants L, m1, and m2 such that (j1) f(x, u)u − 2F(x, u) ≥ m1|u|2, if |u| ≥ L; (j2) |f(x, u)|σ/|u|σ ≤ m2(f(x, u)u − 2F(x, u)), if If 0 is an eigenvalue of Δ2 + cΔ + a (with Navier boundary condition), assuming (F6), problem (1) has at least one nontrivial solution. Assume that φ ∈ C1(X, R) satisfies the Cerami condition (C), φ(−u) = φ(u).
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