Abstract
The existence of multiple solutions for a class of fourth-order elliptic equations with respect to the non-uniformly asymptotically linear conditions is established by using the minimax method and Morse theory.
Highlights
Let H = H ( ) ∩ H ( ) be a Hilbert space equipped with the inner product (u, v)H = ( u v + ∇u∇v) dx, and the deduced norm u H = | u| dx + |∇u| dx.Let λk (k =, . . .) denote the eigenvalues and φk (k =, . . .) the corresponding eigenfunctions of the eigenvalue problem⎧ ⎨– u = λu, in,⎩u =, on ∂, where each eigenvalue λk is repeated as the multiplicity; recall that < λ < λ ≤ λ ≤ · · · ≤ λk → ∞ and that φ (x) > for x ∈
1 Introduction Let H = H ( ) ∩ H ( ) be a Hilbert space equipped with the inner product (u, v)H = ( u v + ∇u∇v) dx, and the deduced norm u
We can observe that k = λk(λk – c), k =, . . . , are eigenvalues of the eigenvalue problem
Summary
We study the problem in the case that f may be non-uniformly asymptotically linear. These new aspects with p-Laplacian were first presented by Duc and Huy in [ ]. (iii) For any sequence {um} converging weakly to u in H, there exists a measurable function g on and a subsequence {umk } of {um} having the following properties: |umk | ≤ g for a.e. x ∈ , for any k and. By condition (H )(iii), there exist measurable functions g , g , v, and a strictly increasing sequence {nk} of positive integer numbers such that Wgi is integrable and lim k→∞.
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