Abstract
The existence of multiple solutions for a class of fourth elliptic equation with respect to the resonance and nonresonance conditions is established by using the minimax method and Morse theory.
Highlights
Let 0 < λ1 < λ2 < · · · < λk < · · · be the eigenvalues of Δ2, H2 Ω ∩ H01 Ω and φ1 x > 0 be the eigenfunction corresponding to λ1
If l in the above condition H2 is an eigenvalue of Δ2, H2 Ω ∩ H01 Ω, problem 1.1 is called resonance at infinity
It is well known that the condition AR plays an important role in verifying that the functional I has a “mountain pass” geometry and a related P S c sequence is bounded in H2 Ω ∩ H01 Ω when one uses the mountain pass theorem
Summary
Consider the following Navier boundary value problem: Δ2u x f x, u , in Ω, 1.1 u Δu 0 in ∂Ω, where Ω is a bounded smooth domain in RN N > 4 , and f x, t satisfies the following: H1 f ∈ C1 Ω × R, R , f x, 0 0, f x, t t ≥ 0 for all x ∈ Ω, t ∈ R; H2 lim|t| → 0 f x, t /t f0, lim|t| → ∞ f x, t /t l uniformly for x ∈ Ω, where f0 and l are constants; H3 lim|t| → ∞ f x, t t − 2F x, t−∞, where F x, t t 0 f x, s ds.Boundary Value ProblemsIn view of the condition H2 , problem 1.1 is called asymptotically linear at both zero and infinity. Under the condition H2 , the critical points of I are solutions of problem 1.1 . If l in the above condition H2 is an eigenvalue of Δ2, H2 Ω ∩ H01 Ω , problem 1.1 is called resonance at infinity.
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